To find the area between two z-scores on a calculator, use the normalcdf command.
What is a z-score?In Mathematics and Geometry, a z-score is also known as a standard score and it's a measure of the distance between a raw score and the mean, when standard deviation units are used.
In Mathematics and Geometry, the z-score of a given sample size or data set can be calculated by using this formula:
Z-score, z = (x - μ)/σ
Where:
σ represents the standard deviation.x represents the sample score.μ represents the mean score.In order to determine the area between two z-scores on a scientific calculator, you should make use of the normalcdf command.
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Suppose that we have 100 apples. In order to determine the integrity of the entire batch of apples, we carefully examine n randomly-chosen apples; if any of the apples is rotten, the whole batch of apples is discarded. Suppose that 50 of the apples are rotten, but we do not know this during the inspection process.
(a) Calculate the probability that the whole batch is discarded for n = 1, 2, 3, 4, 5, 6
(b) Find all values of n for which the probability of discarding the whole batch of apples is at least 99% = 99/100
(a) To calculate the probability that the whole batch is discarded for a given value of n, we need to consider the probability that at least one of the randomly chosen apples is rotten.
Let's calculate this probability for each value of n:
For n = 1:
The probability that at least one apple is rotten is 50/100 = 1/2.
Therefore, the probability that the whole batch is discarded is 1/2.
For n = 2:
The probability that both apples are not rotten is (50/100) * (49/99) = 2450/9900.
Therefore, the probability that at least one apple is rotten is 1 - (2450/9900) = 7450/9900.
Therefore, the probability that the whole batch is discarded is 7450/9900.
For n = 3:
The probability that all three apples are not rotten is (50/100) * (49/99) * (48/98) = 117600/485100.
Therefore, the probability that at least one apple is rotten is 1 - (117600/485100) = 367500/485100.
Therefore, the probability that the whole batch is discarded is 367500/485100.
For n = 4:
The probability that all four apples are not rotten is (50/100) * (49/99) * (48/98) * (47/97) = 342200/1088433.
Therefore, the probability that at least one apple is rotten is 1 - (342200/1088433) = 746233/1088433.
Therefore, the probability that the whole batch is discarded is 746233/1088433.
For n = 5:
The probability that all five apples are not rotten is (50/100) * (49/99) * (48/98) * (47/97) * (46/96) = 50702400/182530530.
Therefore, the probability that at least one apple is rotten is 1 - (50702400/182530530) = 131828130/182530530.
Therefore, the probability that the whole batch is discarded is 131828130/182530530.
For n = 6:
The probability that all six apples are not rotten is (50/100) * (49/99) * (48/98) * (47/97) * (46/96) * (45/95) = 386914800/1251677705.
Therefore, the probability that at least one apple is rotten is 1 - (386914800/1251677705) = 864762905/1251677705.
Therefore, the probability that the whole batch is discarded is 864762905/1251677705.
(b) To find the values of n for which the probability of discarding the whole batch of apples is at least 99/100, we need to find the smallest value of n such that the probability exceeds or equals 99/100.
Starting from n = 1, we can calculate the probability for each value of n until we reach a probability greater than or equal to 99/100:
For n = 1: Probability = 1/2.
For n = 2: Probability = 7450/9900.
For n = 3: Probability = 367500/485100.
For n = 4: Probability = 746233/1088433.
For n = 5: Probability = 131828130/182530530.
For n = 6: Probability = 864762905/1251677705.
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Diversifying an investment portfolio increases the return to
risk ratio. Diversifying internationally heightens the benefits of
diversification. Explain why this is. Diversifying into frontier
and eme
A diversified-portfolio is important because of risk-reduction, smoother-returns, exploiting different opportunities, and risk-allocation.
A "Diversified-Portfolio" refers to an investment portfolio that contains a mix of different asset classes, industries, regions, and securities.
A diversified portfolio is important for several reasons, which are :
(i) Risk-reduction: Diversification helps to reduce the overall risk of investment portfolio. By spreading the investments across different asset classes, industries, regions, and securities, we can mitigate the impact of any individual investment performing poorly.
(ii) Smoother-returns: Diversification can lead to more stable and smoother investment returns over time. Different asset classes or investments tend to perform differently under various market conditions.
(iii) Exploiting different opportunities: By diversifying your portfolio, you can participate in various growth areas and potentially benefit from different economic cycles.
(iv) Risk-allocation: Diversification allows us to allocate the investment capital across different risk profiles based on your investment objectives and risk tolerance.
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The given question is incomplete, the complete question is
Why is it important to have a diversified portfolio?
o make a specific prediction for an individual's score on a given variable, when we know the individual's score on two or more correlated variables, we would use what statistical technique? a. Linear regression b. Multiple correlation coefficient c. Pearson's r correlation coefficient d. Multiple regression
When we want to make a specific prediction for an individual's score on a given variable, when we know the individual's score on two or more correlated variables, we would use the statistical technique known as Multiple Regression.
Multiple Regression is a statistical technique used to assess the relationship between a dependent variable and one or more independent variables. It is used when we need to understand how the value of the dependent variable changes with changes in one or more independent variables. Multiple regression is used when we want to predict a continuous dependent variable from a number of independent variables. In multiple regression, we are interested in the regression equation that uses one or more independent variables to predict a dependent variable. The conclusion of a multiple regression analysis provides information about the relationship between the dependent variable and the independent variables. It tells us whether the relationship is statistically significant, the strength of the relationship, and the direction of the relationship.
Thus, the correct option is (d) Multiple Regression.
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A medical researcher believes that the variance of total cholesterol levels in men is greater than the variance of total cholesterol levels in women. The sample variance for a random sample of 9 men’s cholesterol levels, measured in mgdL, is 287. The sample variance for a random sample of 8 women is 88. Assume that both population distributions are approximately normal and test the researcher’s claim using a 0.10 level of significance. Does the evidence support the researcher’s belief? Let men's total cholesterol levels be Population 1 and let women's total cholesterol levels be Population 2.
1 State the null and alternative hypotheses for the test. Fill in the blank below. H0Ha: σ21=σ22: σ21⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯σ22
2. What is the test statistic?
3. Draw a conclusion
The null and alternative hypotheses for the test are as follows: Null hypothesis (H 0): The variance of total cholesterol levels in men is equal to the variance of total cholesterol levels in women.
Alternative hypothesis (H a): The variance of total cholesterol levels in men is greater than the variance of total cholesterol levels in women.
The null hypothesis states that the variances of total cholesterol levels in men and women are equal, while the alternative hypothesis suggests that the variance in men is greater than that in women. The notation σ21 represents the variance of men's total cholesterol levels, and σ22 represents the variance of women's total cholesterol levels.
The test statistic for comparing variances is the F statistic, calculated as the ratio of the sample variances: F = (sample variance of men) / (sample variance of women). In this case, the sample variance of men is 287 and the sample variance of women is 88.
To draw a conclusion, we compare the calculated F statistic with the critical value from the F distribution at a significance level of 0.10. If the calculated F statistic is greater than the critical value, we reject the null hypothesis and conclude that there is evidence to support the researcher's belief that the variance of total cholesterol levels in men is greater than in women. If the calculated F statistic is not greater than the critical value, we fail to reject the null hypothesis and do not have sufficient evidence to support the researcher's belief.
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A patient needs 3 L of D5W with 20 meq of potassium chloride to infuse over one day (24 hours). The DF is 15 gtt/mL. What is the correct rate of flow in gtt/min? Round to the nearest whole number.
The correct rate of flow in gtt/min for infusing 3 L of D5W with 20 meq of potassium chloride over 24 hours is 31 gtt/min.
To determine the rate of flow in gtt/min, we need to calculate the total number of drops needed over the infusion period and then divide it by the total time in minutes.
First, we need to find the total volume of the solution in milliliters (mL):
3 L = 3000 mL
Next, we calculate the total number of drops needed. We can use the drop factor (DF) of 15 gtt/mL:
Total drops = Volume (mL) x DF
Total drops = 3000 mL x 15 gtt/mL
Next, we calculate the total time in minutes:
24 hours = 24 x 60 minutes = 1440 minutes
Finally, we divide the total drops by the total time in minutes to find the rate of flow in gtt/min:
Rate of flow (gtt/min) = Total drops / Total time (minutes)
Rate of flow (gtt/min) = (3000 mL x 15 gtt/mL) / 1440 minutes
Simplifying the expression, we have:
Rate of flow (gtt/min) ≈ 31.25 gtt/min
Rounding to the nearest whole number, the correct rate of flow in gtt/min is approximately 31 gtt/min.
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question 8 and 9
8- f(t)=e³¹ cos2t 9- f(t)=3+e²2¹-sinh 5t 10- f(t)=ty'.
The integral of f(t) = e³¹ cos²t is (1/2)e³¹t + (1/4)e³¹sin(2t) + C. The integration involves using the power-reducing formula for cosine squared and the substitution method.
The integral of f(t) = e³¹ cos²t is (1/2)e³¹t + (1/4)e³¹sin(2t) + C. To know more about the integration of exponential functions and trigonometric functions, refer here: [link to a reliable mathematical resource].
To integrate f(t) = e³¹ cos²t, we can use the power-reducing formula for cosine squared:
cos²t = (1/2)(1 + cos(2t))
Now, we can rewrite the integral as:
∫ e³¹ cos²t dt = ∫ e³¹ (1/2)(1 + cos(2t)) dt
Distribute e³¹ throughout the integral:
= (1/2) ∫ e³¹ dt + (1/2) ∫ e³¹ cos(2t) dt
Integrating e³¹ with respect to t gives:
= (1/2) e³¹t + (1/2) ∫ e³¹ cos(2t) dt
To integrate ∫ e³¹ cos(2t) dt, we can use the substitution method. Let u = 2t, then du = 2 dt:
= (1/2) e³¹t + (1/4) ∫ e³¹ cos(u) du
Integrating e³¹ cos(u) du gives:
= (1/2) e³¹t + (1/4) e³¹sin(u) + C
Substituting back u = 2t:
= (1/2) e³¹t + (1/4) e³¹sin(2t) + C
Therefore, the integral of f(t) = e³¹ cos²t is (1/2)e³¹t + (1/4)e³¹sin(2t) + C.
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$800 is invested at a rate of 4% and is compounded monthly. Find the balance after 10 years.
The balance after 10 years would be approximately $1,190.96.
To calculate the balance after 10 years of investing $800 at a rate of 4% compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final balance
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
In this case, we have:
P = $800
r = 4% = 0.04 (as a decimal)
n = 12 (compounded monthly)
t = 10 years
Plugging the values into the formula, we have:
A = 800(1 + 0.04/12)^(12 × 10)
Simplifying the calculation inside the parentheses:
A = 800(1 + 0.003333)^120
Using a calculator, we can evaluate (1 + 0.003333)^120 ≈ 1.4887.
A = 800 × 1.4887 ≈ $1,190.96
Therefore, the balance after 10 years would be approximately $1,190.96.
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Question 18 1 points Save An Which of the following statement is correct about the brands and bound algorithm derived in the lectures to solve the max cliquer problem The algorithm is better than bruteforce enumeration because its complexity is subexponential o White the algorithm is not better than tre force enameration tas both have exponential comploty, it can more often as in general do not require the explide construction of all the feasible solutions to the problem The algorithms morient than the force enumeration under no circumstances will construct the set of fantiles
The correct statement about the brands and bound algorithm derived in the lectures to solve the max cliquer problem is that it is not better than brute force enumeration in terms of worst-case time complexity, as both have exponential complexity.
However, the algorithm is more efficient than brute force enumeration in practice as it does not require the explicit construction of all feasible solutions to the problem. The brands and bound algorithm is a heuristic approach that tries to eliminate parts of the search space that are guaranteed not to contain the optimal solution. This means that the algorithm can often find the solution much faster than brute force enumeration. Additionally, the algorithm does not construct the set of cliques/families under any circumstances, which reduces the memory usage of the algorithm.
Overall, while the brands and bound algorithm may not be the most efficient algorithm for solving the max cliquer problem in theory, it is a practical and useful approach for solving the problem in real-world scenarios.
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The cost of producing 6000 face masks is $25,600 and the cost of producing 6500 face masks is $25.775. Use this information to create a function C (a) that represents the cost in dollars a company spends to manufacture x thousand face masks during a month. The linear equation is: C (x) = ____________
The vertical intercept for this graph is at the point ____________ (type a point) and represents a cost of $ ___________when a quantity of _________face masks are produced. The rate of change for C(a) is __________and means the cost is Based on this model, C(11) = ________ which means that when a quantity of ____________ face marks are produced, there is a cost of $ _________
Solving C (a)= 90, 700 shows x = ___________ which represents that for a cost of $. you can produce _____ face masks The appropriate domain of this function is ________ (interval notation- use INF for infinity if needed).
Use the Golden Search method to maximize the following unimodal function, ƒ(X) = −(x − 3)², 2 ≤ x ≤ 4 with A = 0.05.
We will use the Golden Section Search method to maximize the unimodal function ƒ(x) = -(x - 3)² within the interval 2 ≤ x ≤ 4, with an accuracy level of A = 0.05.
The Golden Section Search is an optimization algorithm that narrows down the search interval iteratively by dividing it in a specific ratio based on the golden ratio. In each iteration, we evaluate the function at two points within the interval and compare the function values to determine the new search interval.
To apply the Golden Section Search, we start with the initial interval [a, b] = [2, 4]. The interval is divided into two subintervals based on the golden ratio, giving us two points x₁ and x₂. We evaluate the function at these points and compare the function values to determine the new search interval.
In the first iteration, we evaluate ƒ(x₁) and ƒ(x₂) and compare the values. Since we want to maximize the function, if ƒ(x₁) > ƒ(x₂), we update the search interval to [a, x₂], otherwise, we update it to [x₁, b]. We continue this process iteratively, narrowing down the interval until we reach the desired accuracy level.
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Determine the volume generated of the area bounded by y=√x and y= ½ x rotated around the y-axis.
a. (64/5)π
b. (8/15)π
c. (128/25)π
d. (64/15)
To determine the volume generated by rotating the area bounded by the curves y = √x and y = ½x around the y-axis, we can use the method of cylindrical shells. By setting up the integral and evaluating it, we find that the volume is equal to (64/15)π.
To find the volume, we use the method of cylindrical shells, which involves integrating the circumference of the shells multiplied by their heights. In this case, the height of each shell is the difference between the y-values of the two curves: (√x - ½x).
We integrate with respect to x from the lower bound to the upper bound, which are the x-values where the two curves intersect: x = 0 and x = 4.
Setting up the integral and evaluating it, we find that the volume is equal to ∫(0 to 4) 2πx(√x - ½x) dx. This simplifies to (64/15)π, which is the final answer.
Therefore, the volume generated by rotating the area bounded by the curves y = √x and y = ½x around the y-axis is (64/15)π.
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Consider the vector field F(x, y) = (-2xy, x² ) and the region R bounded by y = 0 and y = x(2-x)
(a) Compute the two-dimensional divergence of the field.
(b) Sketch the region
(c) Evaluate BOTH integrals in Green's Theorem (Flux Form) and verify that both computations match.
The given vector field F(x, y) = (-2xy, x²) is considered along with the region R bounded by y = 0 and y = x(2-x). The two-dimensional divergence of the field is computed.
(a) The two-dimensional divergence of the field F(x, y) = (-2xy, x²) is computed by taking the partial derivative of the first component with respect to x and the partial derivative of the second component with respect to y. The divergence is obtained as -2x.
(b) The region R bounded by y = 0 and y = x(2-x) is sketched. This region is the area between the x-axis and the curve y = x(2-x). It is a triangular region in the coordinate plane.
(c) Green's Theorem (Flux Form) is applied to evaluate two integrals. The first integral involves the line integral of the vector field F(x, y) = (-2xy, x²) over the boundary curve of the region R. The second integral involves the double integral of the divergence of F over the region R. Both integrals are computed, and it is verified that the values obtained from both computations match. This verifies the accuracy of Green's Theorem in this context.
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Two different analytical tests can be used to determine the impurity level in steel alloys. Eight specimens are tested using both procedures, and the results are shown in the following tabulation. Is there sufficient evidence to conclude that both tests give the same mean impurity level, using alpha = 0.01? there sufficient evidence to conclude that both tests give the same mean impurity level since the test statistic in the rejection region. Round numeric answer to 2 decimal places. the tolerance is +/-2%
Based on the given data and using a significance level of 0.01, there is sufficient evidence to conclude that both tests do not give the same mean impurity level in steel alloys. The test statistic falls in the rejection region, indicating a significant difference between the means.
To determine if both tests give the same mean impurity level, we can conduct a hypothesis test. The null hypothesis, denoted as H0, assumes that the mean impurity levels from both tests are equal, while the alternative hypothesis, denoted as H1, assumes that the mean impurity levels are not equal.
Using the given data, we calculate the test statistic, which measures the difference between the sample means of the two tests. Since the population standard deviation is unknown, we use a t-distribution and the appropriate degrees of freedom to calculate the critical value.
By comparing the test statistic to the critical value at a significance level of 0.01, we can determine whether to reject or fail to reject the null hypothesis. If the test statistic falls in the rejection region, which is determined by the critical value, we reject the null hypothesis in favor of the alternative hypothesis, indicating a significant difference between the means.
In this case, since the test statistic falls in the rejection region, we have sufficient evidence to conclude that both tests do not give the same mean impurity level in steel alloys at a significance level of 0.01.
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Research was conducted on the weight at birth of children from urban and rural women. The researcher suspects that there is a significant difference in the mean weight at birth of children between urban and rural women. The researcher selects independent random samples of mothers who gave birth from each group and calculates the mean weight at birth of children and standard deviations. The statistics are summarized in the table below. (a) Test whether there is a difference in the mean weight at birth of children between urban and rural women (use 5% significant level). (30 marks) (b) Assume that medical experts commonly believe that on average a new-born baby in urban areas weighs 3.5000 kg. Is it true that the observed mean weight at birth of children from sample urban mothers is greater than the predicted weight? (use 5% significant level). (20 marks)
(a) To test the difference in mean weight at birth between urban and rural women, a two-sample t-test can be used. The significance level of 5% implies that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis.
The t-test compares the means of the two samples, considering their respective sample sizes and standard deviations. By calculating the test statistic and comparing it to the critical value from the t-distribution with appropriate degrees of freedom, we can determine whether the observed difference is statistically significant.
(b) To test whether the observed mean weight at birth of children from sample urban mothers is greater than the predicted weight of 3.5000 kg, a one-sample t-test can be conducted. The null hypothesis (H₀) assumes that the mean weight is equal to or less than 3.5000 kg, while the alternative hypothesis (H₁) suggests that the mean weight is greater.
Similar to the previous test, the t-test calculates the test statistic using the sample mean, standard deviation, and sample size. By comparing the test statistic to the critical value from the t-distribution with appropriate degrees of freedom, we can determine whether the observed mean weight is significantly greater than the predicted weight.
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77. Find the inverse of the nonsingular matrix -4 1 6 -2]
The inverse of the nonsingular matrix [-4 1; 6 -2] is [1/2 1/2; -3/4 -1/4].
To find the inverse of a matrix, we follow a specific procedure. Let's consider the given matrix [-4 1; 6 -2] and find its inverse.
Step 1: Calculate the determinant of the matrix.
The determinant of the matrix is found by multiplying the diagonal elements and subtracting the product of the off-diagonal elements. For the given matrix, the determinant is:
Det([-4 1; 6 -2]) = (-4) * (-2) - (1) * (6) = 8 - 6 = 2.
Step 2: Determine the adjugate matrix.
The adjugate matrix is obtained by taking the transpose of the matrix of cofactors. To find the cofactors, we interchange the signs of the elements and compute the determinants of the remaining 2x2 matrices. For the given matrix, the cofactor matrix is:
[-2 -6; -1 -4].
Taking the transpose of this matrix, we get the adjugate matrix:
[-2 -1; -6 -4].
Step 3: Calculate the inverse matrix.
The inverse of the matrix is obtained by dividing the adjugate matrix by the determinant. For the given matrix, the inverse is:
[1/2 1/2; -3/4 -1/4].
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When doing 2 proportion testing, you must check the Success/Failure Condition. Which of the following statements is true?
I. If both samples pass the success part but do not pass the failure part, it is a violation but does not need to be discussed in the conclusion
II. If one sample passes both parts but the other does not pass either part, it is a violation that needs to be discussed in the conclusion
III. If one sample passes both parts but the other only passes the success part, it is not a violation
IV. If both samples do not pass the success part but pass the failure part, it is a violation that must be discussed in the conclusion
a. II and III
b. I and IV
c. II and IV
The correct statement is: c. II and IV for two proportion testing.
In two proportion testing, the success/failure condition refers to the number of successes and failures in each sample. The condition states that both samples should have a sufficient number of successes and failures for the test to be valid.
II. If one sample passes both parts (has a sufficient number of successes and failures) but the other does not pass either part, it is a violation that needs to be discussed in the conclusion. This is because the sample that does not meet the success/failure condition may affect the validity and reliability of the test results.
IV. If both samples do not pass the success part (do not have a sufficient number of successes) but pass the failure part (have a sufficient number of failures), it is a violation that must be discussed in the conclusion. This violation indicates that the test may not be appropriate for analyzing the proportions in the given samples.
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Write the hypothesis for the following cases:
1- A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles.
2- A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?
The null and alternative hypothesis are significant.
1) Hypothesis is a proposed explanation made on the basis of limited evidence as a starting point for further investigation. For the given case, the hypothesis can be stated as:
Null Hypothesis (H0): The average lifespan of the deluxe tire is greater than or equal to 50,000 miles.
Alternative Hypothesis (Ha): The average lifespan of the deluxe tire is less than 50,000 miles.
2) The null hypothesis states that there is no statistically significant difference between the two groups being tested.
It is often denoted by H0.
The alternative hypothesis is often denoted by Ha and states that there is a statistically significant difference between the two groups being tested.In this case, the null and alternative hypotheses would be:Null Hypothesis (H0):
The population mean time on death row is 15 years.
Alternative Hypothesis (Ha): The population mean time on death row is not 15 years.
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.
2. y^3y'+x^3=0
3. y' = sec62 y
4. y' sin 2πx = πy cos 2πx
5. yy'+36x =0
The given differential equations are:
1. y^3y' + x^3 = 0
2. y' = sec^2(θ) y
3. y' sin(2πx) = πy cos(2πx)
4. yy' + 36x = 0
1. The differential equation y^3y' + x^3 = 0 is a first-order nonlinear differential equation. To solve it, we can separate the variables by rewriting it as y' = -x^3/y^3. Then, we can integrate both sides to obtain the solution.
2. The differential equation y' = sec^2(θ) y is a separable differential equation. We can rewrite it as dy/y = sec^2(θ) dθ. Integrating both sides will give us the solution.
3. The differential equation y' sin(2πx) = πy cos(2πx) is also a separable differential equation. By dividing both sides by y sin(2πx) and integrating, we can find the solution.
4. The differential equation yy' + 36x = 0 is a first-order linear differential equation. It can be solved using the method of integrating factors or by rearranging it as y' = -36x/y and then integrating both sides.
Each of these differential equations requires different techniques to solve, such as separation of variables, integrating factors, or rearranging the equation. The specific solution for each equation will depend on the given initial conditions or any additional constraints provided.
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A binomial distribution is composed of six fixed identical trials and the probability of success is 0,83. Therefore the mean and standard deviation of the binomial distribution are equal to: a. 4.98 and 0.9201 b. 1.02 and 0.9201
c. 1.50 and 0.866 d. 1.50 and 0.980
The binomial distribution that is composed of six identical fixed trials and a success probability of 0.83 has a mean and standard deviation of 4.98 and 0.9201, respectively. The correct option is A
The given probability distribution is a binomial distribution that consists of six identical fixed trials and the probability of success is 0.83.
Using the formula for the mean and standard deviation of the binomial distribution, we can solve this problem.
The formula for the mean and standard deviation is as follows:
Mean (μ) = [tex]n * p[/tex]
= [tex]6 * 0.83[/tex]
= 4.98
Standard deviation (σ) = √(n * p * q)
= √(6 * 0.83 * 0.17)
= 0.9201
Therefore, the mean and standard deviation of the binomial distribution are 4.98 and 0.9201, respectively. Thus, the correct option is (a)
The binomial distribution that is composed of six identical fixed trials and a success probability of 0.83 has a mean and standard deviation of 4.98 and 0.9201, respectively.
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1) Differentiate. a) f(x)= 1 (cos(x5-5x)*
b) f(x) = sin-1(x3 - 3x)
The differentiation of the given functions are as follows; a) [tex]f(x) = 1cos(x5 - 5x) :[/tex]
[tex]df/dx = sin(x^5 - 5x)(5x^4 - 5)b) f(x)[/tex]
[tex]= sin-1(x3 - 3x) :[/tex]
[tex]f′(x) = (3x^2 - 3) / √(1 - (x^3 - 3x)^2).[/tex]
Differentiation of trigonometric functions The process of finding the derivative of a function is called differentiation. In mathematics, differentiation is a primary mathematical concept that has a variety of applications in various fields. It is applied to trigonometric functions as well. The trigonometric functions that are primarily differentiated include sine, cosine, tangent, cotangent, secant, and cosecant. Therefore, the differentiation of the given functions is as follows; a) [tex]f(x) = 1cos(x5 - 5x)[/tex] The given function is
[tex]f(x) = 1cos(x5 - 5x).[/tex] To find its derivative, we use the formula of the chain rule of differentiation:
[tex]`(f(g(x)))′ = f′(g(x))g′(x)`[/tex] Given that,
[tex]`f(x) = 1cos(x5 - 5x)`[/tex] Let
[tex]`u = (x^5 - 5x)`[/tex] So,
[tex]`f(x) = 1cosu`[/tex] Now differentiate `u` with respect to `x` and get `du/dx
[tex]= 5x^4 - 5`[/tex] Then
[tex]`df/dx = -sinu (du/dx)` But `cosu[/tex]
[tex]= cos(x^5 - 5x)`[/tex] Therefore, the differentiation of
[tex]f(x) = 1cos(x5 - 5x)[/tex] is given by
[tex]`df/dx = sin(x^5 - 5x)(5x^4 - 5)`b)[/tex]
[tex]f(x) = sin-1(x3 - 3x).[/tex]
The given function is [tex]f(x) = sin-1(x3 - 3x)[/tex] To find its derivative, we apply the formula of the chain rule of differentiation: [tex]`(f(g(x)))′ = f′(g(x))g′(x)`[/tex] Let
[tex]`u = x^3 - 3x`[/tex] and
[tex]`y = sin-1u`[/tex] Hence,
[tex]`y′ = dy/du * du/dx`[/tex] Differentiate `y` with respect to `u` and get
[tex]`dy/du = 1/√(1 - u^2)`[/tex] Differentiate `u` with respect to `x` and get
[tex]`du/dx = 3x^2 - 3`[/tex] Therefore,
[tex]`y′ = (1/√(1 - u^2)) * (3x^2 - 3) `[/tex] Hence, the differentiation of
[tex]f(x) = sin-1(x3 - 3x)[/tex] is given by
[tex]`f′(x) = (3x^2 - 3) / √(1 - (x^3 - 3x)^2)`[/tex] In conclusion, the differentiation of the given functions are as follows; a)
[tex]f(x) = 1cos(x5 - 5x)[/tex] :
[tex]df/dx = sin(x^5 - 5x)(5x^4 - 5)b) f(x)[/tex]
[tex]= sin-1(x3 - 3x)[/tex] :
[tex]f′(x) = (3x^2 - 3) / √(1 - (x^3 - 3x)^2).[/tex]
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Consider the problem maxx +2y subject to x² + y² ≤ 1 and x + y ≥ 0 a. Write down the first order conditions. b. Solve the problem.
The problem involves maximizing the objective function f(x, y) = x + 2y, subject to the constraints x² + y² ≤ 1 and x + y ≥ 0.
In order to solve the problem, we need to determine the first-order conditions and find the optimal solution.
a. First-order conditions:
To find the first-order conditions, we need to consider the Lagrangian function L(x, y, λ) = f(x, y) - λ(g(x, y)), where g(x, y) represents the constraints. In this case, the constraints are x² + y² ≤ 1 and x + y ≥ 0.
The first-order conditions are:
∂L/∂x = 1 - 2λx = 0
∂L/∂y = 2 - 2λy = 0
g(x, y) = x² + y² - 1 ≤ 0
h(x, y) = -(x + y) ≤ 0
b. Solving the problem:
To solve the problem, we need to solve the first-order conditions and check the feasibility of the constraints.
From the first-order conditions, we have:
1 - 2λx = 0 --> x = 1/(2λ)
2 - 2λy = 0 --> y = 1/(2λ)
Substituting these values into the constraint equations, we have:
(1/(2λ))² + (1/(2λ))² ≤ 1 --> 1/(4λ²) + 1/(4λ²) ≤ 1 --> 1/λ² ≤ 1 --> λ² ≥ 1 --> λ ≥ 1 or λ ≤ -1
Since λ must be non-negative, we have λ ≥ 1.
Substituting λ = 1 into the expressions for x and y, we get:
x = 1/2
y = 1/2
Therefore, the optimal solution is x = 1/2 and y = 1/2, which maximizes the objective function x + 2y subject to the given constraints.
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Set up the definite integral required to find the area of the
region between the graph of y = 11 − x 2 and y = − 25 x + 165 over
the interval − 1 ≤ x ≤ 1
The integral we need to solve is:
[tex]\int\limits^1_{-1} {(- x^2 + 25 x - 154)} \, dx[/tex]
How to find the area between the curves?
Here we just need to integrate the difference between the two curves in the given region, so we will get:
[tex]\int\limits^1_{-1} {11 - x^2 - (-25 x + 165)} \, dx[/tex]
Simplify that to get:
[tex]\int\limits^1_{-1} {(- x^2 + 25 x - 154)} \, dx[/tex]
We will get the area:
area = [ (1/3)*( - (1)^3 - (-1)^3) - 154*(1 - (-1))
area = -308.6
A negative area means that the first function is mostly below the second one.
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Let f(x) = √2x - 10 and the virtual line joining the origin (0, 0) to a point Q moving on the curve of the function f. The curve and the line are shown below. a) Determine the coordinates of point Q that would maximize the viewing angle theta (0) of an observer whose eye, located at the origin, follows the displacement of the point Q along the curve. Note that tan(0) = b) Determine this maximum angle (in degrees)
To determine the coordinates of point Q that would maximize the viewing angle θ(0) and find the maximum angle in degrees, we need to find the maximum value of the tangent function.
Given that f(x) = √(2x) - 10, we want to find the maximum value of tan(θ(0)).
The tangent function is defined as tan(θ) = opposite/adjacent, which in this case is y/x.
Let's find the equation of the line connecting the origin (0, 0) to point Q on the curve.
The equation of the line is y = mx, where m is the slope of the line.
The slope, m, is given by m = (f(x) - 0)/(x - 0) = f(x)/x.
Substituting f(x) = √(2x) - 10, we have m = (√(2x) - 10)/x.
Now, let's substitute y = mx into the equation of the curve:
√(2x) - 10 = (√(2x) - 10)/x * x.
Simplifying, we have:
√(2x) - 10 = (√(2x) - 10).
Both sides of the equation are equal, indicating that any point on the curve satisfies this equation.
To maximize the viewing angle θ(0), we need to find the point Q on the curve where the tangent function tan(θ(0)) is maximized.
The tangent function is maximized when the slope of the line connecting the origin to point Q is maximized. This occurs when the line is tangent to the curve.
To find the point Q where the line is tangent to the curve, we need to find the maximum value of the slope (√(2x) - 10)/x.
Taking the derivative of the slope with respect to x and setting it equal to zero to find the critical points:
d/dx [(√(2x) - 10)/x] = 0.
Using the quotient rule for differentiation, we get:
[(1/2√(2x))x - (√(2x) - 10)]/x^2 = 0.
Simplifying, we have:
(1/2√(2x))x - (√(2x) - 10) = 0.
Solving for x, we find:
x = 20.
Now, we substitute x = 20 into the equation of the line to find the y-coordinate of point Q:
y = (√(2x) - 10) = (√(2*20) - 10) = 0.
Therefore, the coordinates of point Q that maximize the viewing angle θ(0) are (20, 0).
Now, to determine the maximum angle θ(0) in degrees, we can calculate it using the arctan function:
θ(0) = arctan(m) = arctan((√(2x) - 10)/x) = arctan((√(2*20) - 10)/20) ≈ 43.60 degrees.
Therefore, the maximum angle θ(0) is approximately 43.60 degrees.
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Consider the following differential equation.
x dy/dx - y = x2 sin(x)
Find the coefficient function P(x) when the given differential equation is written in the standard form dy/dx + P(X)y= f (x).
P (x)= - ½
Find the integrating factor for the differential equation.
E(P(x) dx = 1/3
Find the general solution of the given differential equation.
y(x) = x sin(x) x2cos(x) + Cx
Give the largest interval over which the general solution is defined. (Think about the implications of any singular points. Enter your answer using interval notation.)
Determine whether there are any transient terms in the general solution. (Enter the transient terms as a comma-separated list; if there are none, enter NONE.)
Given: differential equation is x dy/dx - y = x^2 sin(x)
The standard form of the differential equation is dy/dx + P(x)y = f(x)
Here, P(x) is the coefficient function and f(x) = x^2 sin(x).
We can write the given differential equation as (x d/dx - 1)y = x^2 sin(x)
Comparing this with the standard form, we getP(x) = -1/x
The integrating factor for the differential equation is given by e^(integral(P(x) dx))
So, e^(integral(P(x) dx)) = e^(integral(-1/x dx)) = e^(-ln(x)) = 1/x
The integrating factor for the given differential equation is 1/x.
Given differential equation is x dy/dx - y = x^2 sin(x)
Rearranging, we getx dy/dx - y/x = x sin(x)
Differentiating with respect to x, we getd/dx(xy) - y = x sin(x) dx
Multiplying both sides by the integrating factor 1/x, we getd/dx((xy)/x) = sin(x) dx
Integrating both sides with respect to x, we getxy = -cos(x) + Cx
Taking y to one side, we gety(x) = x sin(x) x^2 cos(x) + Cx
Thus, the general solution of the given differential equation is y(x) = x sin(x) x^2 cos(x) + Cx
Give the largest interval over which the general solution is defined.
The given solution is defined for all x, except x=0.
Therefore, the largest interval over which the general solution is defined is (-∞, 0) U (0, ∞).
Determine whether there are any transient terms in the general solution.
There are no transient terms in the general solution.
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A mean weight of 500 sample cars found (1000 + B) Kg. Can it be reasonably regarded as a sample from a large population of cars with mean weight 1500 Kg and standard deviation 130 Kg?
Test at 5% level of significance. (20 Marks) B = 022
The data suggests that the sample of 500 cars does not come from a population with a mean weight of 1500 Kg at a 5% level of significance.
To determine if the sample of 500 cars can be reasonably regarded as a sample from a population with a mean weight of 1500 Kg and a standard deviation of 130 Kg, we can perform a hypothesis test.
Let's set up the null and alternative hypotheses:
Null hypothesis (H0): The sample is from a population with a mean weight of 1500 Kg.
Alternative hypothesis (Ha): The sample is not from a population with a mean weight of 1500 Kg.
We can conduct a one-sample t-test to test this hypothesis. The test statistic is calculated as:
t = ([tex]\bar X[/tex] - μ) / (s / √n)
Where:
[tex]\bar X[/tex] is the sample mean weight (1000 + B)
μ is the population mean weight (1500)
s is the sample standard deviation (unknown)
n is the sample size (500)
We are given that B = 022, so the sample mean weight can be calculated as:
[tex]\bar X[/tex] = 1000 + B = 1000 + 0.022 = 1000.022 Kg
Since the sample standard deviation is unknown, we cannot directly calculate the test statistic. However, if the sample size is sufficiently large (usually considered when n > 30), we can assume that the sample standard deviation is a good estimate of the population standard deviation.
Given that we have a large sample size of 500, we can proceed with the assumption that the sample standard deviation is a good estimate of the population standard deviation (130 Kg).
Next, we calculate the t-value using the formula above and the given values:
t = (1000.022 - 1500) / (130 / √500)
Using a statistical calculator or software, we can find the critical t-value at a 5% level of significance with 499 degrees of freedom (500 - 1). The critical t-value for a one-tailed test is approximately 1.646.
If the calculated t-value is greater than the critical t-value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Calculate the t-value:
t = (1000.022 - 1500) / (130 / √500) ≈ -31.3
Since the calculated t-value (-31.3) is much smaller than the critical t-value (1.646), we reject the null hypothesis. Therefore, the sample cannot be reasonably regarded as a sample from a population with a mean weight of 1500 Kg.
In conclusion, the data suggests that the sample of 500 cars does not come from a population with a mean weight of 1500 Kg at a 5% level of significance.
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Given f(x, y) = 2y² + xy³ + 2ex, find f_xx
a. 2e^x
b. 3e^x
c. e^x
d. 6e^x
The solution to find f_xx is to take the second partial derivative of f(x, y) with respect to x, holding y constant. This gives f_xx = 2e^x.
To find f_xx, we first need to find the partial derivative of f(x, y) with respect to x. This gives f_x = y^3 + 2e^x.
Then, we take the partial derivative of f_x with respect to x. This gives f_xx = 2e^x.
Therefore, the answer is a. 2e^x.
Here is a more information of the steps involved:
1. We start by finding the partial derivative of f(x, y) with respect to x. This gives f_x = y^3 + 2e^x.
2. Then, we take the partial derivative of f_x with respect to x. This gives f_xx = 2e^x.
3. Therefore, the answer is a. 2e^x.
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(a) Let f(x) = x² + 1. Compute f(0), ƒ(-1), f(1/2), and f(√2).
(b) For what values of x is it true that
(i) f(x) = f(-x)?
(ii) f(x + 1) = f(x) + f(1)?
(iii) f(2x) = 2ƒf(x)?
Problem 2
The cost of producing x units of a commodity is given by C(x) = 1000 + 300x + x².
(a) Compute C(0), C(100), and C(101) - C(100).
(b) Compute C(x + 1) - C(x), and explain in words the meaning of the difference.
For problem 1,
we are given f(x) = x² + 1.
The
values
of f(0), f(-1), f(1/2), and f(√2) are 1, 2, 1.25, and 3, respectively.
For problem 2,
We are given C(x) = 1000 + 300x + x².
The
marginal cost
is constant at 300.
We are given f(x) = x² + 1
Let’s compute the values of x for which the following hold true:
(i) f(x) = f(-x)
x² + 1 = (-x)² + 1 x²
=x²
Therefore, the above holds true for all x.
(ii) f(x + 1) = f(x) + f(1) (x + 1)² + 1
=x² + 1 + 1² + 1 x² + 2x + 1 + 1
= x² + 2 2x
= 0 x
= 0
Therefore, the above holds true only for x = 0.
(iii) f(2x) = 2f(x) (2x)² + 1
= 2(x² + 1) 4x² + 1
= 2x² + 2 2x²
= 1 x
= ± 1/√2
Therefore, the above holds true for x = 1/√2 and
x = -1/√2
(i) f(x) = f(-x) holds
true
for all x.
(ii) f(x + 1)
= f(x) + f(1) holds true only for
x = 0.
(iii) f(2x) = 2f(x) holds true for
x = 1/√2 and
x = -1/√2.
We are given C(x) = 1000 + 300x + x².
C(x + 1) – C(x) = [1000 + 300(x + 1) + (x + 1)²] – [1000 + 300x + x²] C(x + 1) – C(x)
= 300 + 2x
The above difference gives the marginal cost of producing one extra unit of the
commodity
.
The marginal cost is a constant value of 300, whereas, 2x is the variable cost associated with the
production
of an additional unit of the commodity.
C(x + 1) – C(x) gives the marginal cost of producing one extra unit of the commodity.
The marginal cost is constant at 300, whereas 2x is the variable cost associated with the production of an additional unit of the commodity.
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С x 4 Gx 2 + y2 = Being the curre from Point (30) to point Co-3) on the circle 9 as sy 2 ds c Calculate the Integral)
The solution of the integral is ∫[C] (c * 4 * G * 2 + y * 2) ds = ∫[arcsin(-1/3), arccos(1/3)] (c * 4 * G * 2 + 9 * sin(t)²) * 9 dt
To calculate the integral of the given expression over the curve on the circle, we first need to parameterize the curve. Let's denote the parameter along the curve as t. We can represent the curve on the circle as (x(t), y(t)), where x(t) and y(t) are the x-coordinate and y-coordinate of the curve at parameter t.
Since the curve lies on the circle with center C and radius 9, we can use the equation of a circle to find x(t) and y(t). The equation of a circle with center (a,b) and radius r is given by:
(x - a)² + (y - b)² = r²
In our case, the center C is (0,0) and the radius is 9. Plugging in these values, we have:
x(t)² + y(t)² = 9²
Next, let's solve for x(t) and y(t) in terms of the parameter t. One way to parameterize the curve on the circle is by using trigonometric functions. We can express x(t) and y(t) as:
x(t) = 9 * cos(t) y(t) = 9 * sin(t)
Now that we have the parameterization of the curve, we can calculate the line integral. The line integral of a function f(x, y) over a curve C parameterized by x(t) and y(t) is given by:
∫[C] f(x, y) ds = ∫[a,b] f(x(t), y(t)) * ||r'(t)|| dt
In this case, the function we want to integrate is c * 4 * G * 2 + y * 2, where c and G are constants. Plugging in the parameterization of the curve, we have:
∫[C] (c * 4 * G * 2 + y * 2) ds = ∫[a,b] (c * 4 * G * 2 + 9 * sin(t)²) * ||r'(t)|| dt
To calculate ||r'(t)||, we differentiate x(t) and y(t) with respect to t:
x'(t) = -9 * sin(t) y'(t) = 9 * cos(t)
The magnitude of the derivative vector r'(t) is given by ||r'(t)|| = √(x'(t)² + y'(t)²). Plugging in the values, we have:
||r'(t)|| = √((-9 * sin(t))² + (9 * cos(t))²) = √(81 * sin(t)² + 81 * cos(t)²) = √(81) = 9
Therefore, the line integral simplifies to:
∫[C] (c * 4 * G * 2 + y * 2) ds = ∫[a,b] (c * 4 * G * 2 + 9 * sin(t)²) * 9 dt
Now, we need to determine the limits of integration. We are given that the curve starts at point (3,0) and ends at point (0,-3). We can find the values of t that correspond to these points by plugging the values of x and y into the parameterization equations:
When x = 3 and y = 0: 3 = 9 * cos(t) => cos(t) = 1/3 => t = arccos(1/3)
When x = 0 and y = -3: -3 = 9 * sin(t) => sin(t) = -1/3 => t = arcsin(-1/3)
Therefore, the limits of integration are a = arcsin(-1/3) and b = arccos(1/3).
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If A and B are square matrices of order 3 and 2A^-1B = B - 4I,
show that A - 2I is invertible.
Given that the two matrices A and B are square matrices of order 3 and 2 respectively and, 2A⁻¹B = B - 4I. To show that A - 2I is invertible, we need to prove that det(A - 2I) ≠ 0.The equation given can be written as:2A⁻¹B = B - 4I2A⁻¹B + 4I = B2(A⁻¹B + 2I) = B
Here, B can be replaced by 2(A⁻¹B + 2I) which gives:B = 2(A⁻¹B + 2I)Now, the equation can be written as:A⁻¹B = ½(B - 4I)Now, we have two matrices, A and B, where A is a square matrix of order 3 and B is a square matrix of order 2.Given, 2A⁻¹B = B - 4I2(A⁻¹B) + 4I = BSubstituting ½(B - 4I) for A⁻¹B,
we get:2 * ½(B - 4I)A = ½(B - 4I)A = ¼(B - 4I)We know that A is a square matrix of order 3 and A - 2I is invertible, i.e. (A - 2I)⁻¹ exists. Let's assume that det(A - 2I) = 0, which means (A - 2I)⁻¹ does not exist.Therefore, det(A - 2I) ≠ 0 and (A - 2I)⁻¹ exists. So, A - 2I is invertible and the proof is complete.
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Given matrices A and B are square matrices of orders 3 and 2 respectively and 2A^−1B = B - 4I, we have to show that A - 2I is invertible.
Now, if (2A^−1 - I) is invertible, then we can write it as(2A^−1 - I)^-1 = 1/2 A(B)^-1If we multiply both sides of the equation with B, we get: B (2A^−1 - I) (1/2 A(B)^-1) = -2I(B)^-1By distributive property, it becomes:
B [(2A^-1 × 1/2A(B)^-1) - (I × 1/2A(B)^-1)] = -2I(B)^-1Let us simplify[tex]2A^-1 × 1/2A(B)^-1 = BB(B)^-1 =[/tex] I, so the equation becomes:
B (I - 1/2(B)^-1) = -2I(B)^-1Or, B [I - 1/2(B)^-1] = -2I(B)^-1Thus, (I - 1/2(B)^-1) is invertible. Thus, the matrices 2A^−1 - I and I - 1/2(B)^-1 are invertible.
As the product of two invertible matrices is also invertible, the matrix B (2A^−1 - I) (1/2 A(B)^-1) is invertible.
Now, A - 2I = (1/2)A [2A^−1 × B - 2I]Thus, we get:
A - 2I = (1/2)A [B (2A^−1 - I) (1/2 A(B)^-1) - 2I]Now, we know that the product of invertible matrices is invertible.
So,[tex]B (2A^−1 - I) (1/2 A(B)^-1[/tex]) is invertible. And so, [tex](B (2A^−1 - I) (1/2 A(B)^-1) - 2I)[/tex]is also invertible. Finally, (1/2)A [B (2A^−1 - I) (1/2 A(B)^-1) - 2I] is invertible.So, A - 2I is invertible. Hence, this is the required proof and we have shown that A - 2I is invertible.
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pls clear hand writing
a) The sum of the first n terms of the progression 36,34,32, ...is 0. Find n and the tenth (4 marks) term.
n = 37, and tenth term = 18
Given progression,
36, 34, 32, ...
The sum of the first n terms is 0
First term(a1) = 36
The common difference (d)= 34-36 = -2,
The formula of the sum of the first n term is,
[tex]Sn = \frac{n}{2} [2a_{1} + (n - 1)d][/tex]
substitue the values Sn= 0, a1= 36, d= -2 in the above equation to find n
[tex]0[/tex]= [tex]\frac{n}{2} [2(36) + (n-1) (-2)][/tex]
[tex]0 = \frac{n}{2}[72- 2n+ 2][/tex]
[tex]0 = \frac{n}{2}[74 - 2n][/tex]
[tex]74 - 2n = 0[/tex]
[tex]2n = 74[/tex]
[tex]n = \frac{74}{2}[/tex]
[tex]n = 37[/tex]
n = 37
The formula for finding the nth term(10th term):
[tex]a_{n} = a1 + (n - 1)d[/tex]
n = 10, a1 = 36, d = -2
[tex]a_{10} = 36 + (10-1)(-2)[/tex]
[tex]a_{10} = 36 + 9(-2)[/tex]
[tex]a_{10} = 36 - 18[/tex]
[tex]a_{10} = 18[/tex]
[tex]a_{10}[/tex] = 18
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