The values in the table represent a linear function. What is the common difference of the associated arithmetic sequence?


x y
1 6
2 22
3 38
4 54
5 70


answer choices: A.16
B. 20
C.1
D.5

As x approaches 0, the limit lim 1-0 sin² (161) 1 evaluates to sin² (161).

To find the limit, we must determine what happens to the function as x approaches 0. A limit is a value that a function approaches as x approaches a particular point or infinity. We can calculate a limit by analyzing the function's behavior as x approaches the limiting value from the left or right.

It can be simplified as follows;

Lim 1-0 sin² (161) 1

= lim 1-0 sin² (161) * lim 1-0 1 [using product rule]

= sin² (161) * 1

= sin² (161)

Therefore, this means that as x approaches 0, the limit of the function sin² (161) 1 approaches the value of sin² (161).

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Given that the number x of bicycle helmets a retail chain is willing to sell per week at a price of p is given by x=81\sqrt{p}+9-350 for 20\le p\le 100. The equilibrium price of bicycle is $43.58 and the equilibrium quantity of helmets is 1179 helmets.

The demand curve for the given equation is:

x = 81√p + 9 - 350

The supply curve is a straight line, and it's given by

y = 50p - 550.

Plot the demand curve in the same graph as the supply curve for 20 ≤ p ≤ 100.

Also, draw horizontal lines representing the equilibrium quantity of helmets.

At the equilibrium price, supply equals demand.800-700-600-500-

Part (a)The equilibrium price for bicycles

The equilibrium price is the one where the quantity demanded and the quantity supplied are the same.

Equating the given supply equation to the demand equation:

50p - 550

= 81√p + 9 - 350

Simplifying this equation:

31p - 191 = 81√p

Taking the square of both sides of the equation:

961p² - 116411p + 36481 = 0

On solving this quadratic equation we get:

p = 43.58 or

p = 87.88

As per the given interval, $20 \le p \le 100$,

so the equilibrium price is

p = 43.58

Part (b)The equilibrium quantity of helmets

Substitute $p

= 43.58$ into the supply equation to get the equilibrium quantity:

q = 50(43.58) - 550q

= 1179

Therefore, the equilibrium quantity of helmets is 1179 helmets.

Hence, the solution is:

The equilibrium price of bicycle is $43.58 and the equilibrium quantity of helmets is 1179 helmets.

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The applicant should spend $50 on the preparatory course as it doubles his probability of achieving a score above 750, increasing his chances of getting into a top 20 MBA program.

To determine whether the applicant should spend $50 on the preparatory course, we need to evaluate the probability tree.

The tree diagram would consist of two branches: one representing the applicant not taking the course and the other representing the applicant taking the course.

Starting with the branch of not taking the course, the probability is -20% (0.2) of achieving a score above 750.

For the branch of taking the course, the probability of achieving a score above 750 would need to double from 0.2 to 0.4, as stated by the applicant.

Considering the increase in probability and the cost of the course, it would be beneficial for the applicant to spend $50 on the preparatory course since the probability of achieving a score above 750 doubles.

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Answer:

1.8*10000=1800g

1800:

In order to find the maximum profit for the company that manufactures and sells Q items per month using the given monthly profit function TP(Q)=Q³-720² +2254Q-2314, we need to use the 3-step optimization process.

The 3-step optimization process includes the following:

Therefore, the maximum profit for the company is $2,397,748.29.

Step 1: Find the critical values for TP(Q)

Step 2: Use the second derivative test to determine whether each critical value corresponds to a maximum, minimum, or neither

Step 3: Compare all values of TP(Q) corresponding to maximums to find the absolute maximum In

Step 1, we need to find the critical values for TP(Q). To do this, we take the first derivative of TP(Q) and set it equal to zero:

TP'(Q)

= 3Q² - 1440Q + 2254

= 0

Solving for Q using the quadratic formula, we get:

Q = [1440 ± sqrt(1440² - 4(3)(2254))] / 2(3)≈ 231.31 or 678.03

These are the critical values of

Step 2:

To determine whether each critical value corresponds to a maximum, minimum, or neither, we use the second derivative test.

Taking the second derivative of TP(Q),

we get: TP''(Q) = 6Q - 1440Plugging in the first critical value,

Q ≈ 231.31, we get: TP''(231.31) ≈ -1657.38

Since the second derivative is negative at this critical value, the function has a relative maximum at Q ≈ 231.31. Plugging in the second critical value, Q ≈ 678.03, we get: TP''(678.03) ≈ 2463.16

Since the second derivative is positive at this critical value, the function has a relative minimum at

Q ≈ 678.03.Step 3: Now we compare all values of TP(Q) corresponding to maximums to find the absolute maximum.

Plugging in the first critical value,

Q ≈ 231.31, into the original function, we get:

TP(231.31)

≈ $1,561,534.83

Plugging in the second critical value,

Q ≈ 678.03, into the original function, we get:

TP(678.03)

≈ $2,397,748.29

Therefore, the maximum profit for the company is $2,397,748.29.

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The equation can be written in the form y = mx + c as y = (3/5)x + 4/5 and the correct option is D, i.e., m = 0.6, c = 0.8.

The linear equation 5y - 3x -4 = 0 can be written in the form y m and c is m = 0.6, c = 0.8. We are given the linear equation 5y - 3x - 4 = 0 which is not in the form y = mx + c. To write this in y = mx + c form, we have to isolate the y variable and rewrite the equation in terms of y.Therefore,

5y - 3x - 4 = 0

⇒ 5y = 3x + 4

⇒ y = (3/5)x + 4/5

Comparing the equation with the standard equation y = mx + c, we get:

m = 3/5 and c = 4/5

Hence, the equation can be written in the form y = mx + c as y = (3/5)x + 4/5 and the correct option is D, i.e., m = 0.6, c = 0.8.

Therefore, we can say that the linear equation 5y - 3x -4 = 0 can be written in the form y m and c as m = 0.6, c = 0.8.

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For the given "regression-model", We get :

(a) The predicted delivery time is : 54.501,

(b) The adjusted R² is 95.64%, and overall model significance at 2.5% level is 264.

Part (a) : The multiple linear-regression model that was fitted is : Y' = 2.341 + 1.616×X₁ + 0.144×X₂,

Predict : We use the regression model by replacing X₁ with 10 and X₂ with 250,

So, Y' = 2.341 + 1.616(10) + 0.144(250),

Y' = 54.501,

So, the predicted delivery time is : 54.501,

Part (b) : The formula to calculate the adjusted R² is : 1 - (1 - R²)×(n - 1)/(n - k - 1),

where k is number of "independent-variables" which is : 2,

So, We get,

= 1 - (1 - 0.96)(25 - 1)/(25 - 2 - 1),

= 1 - 0.0436 ≈ 0.9564

So, the adjusted R² is 95.64%,

The "Overall-Significance" : F-Test statistics is = (R²/k)/{(1 - R²)/(n - k - 1)},

Substituting the values,

we get,

= (0.96/2)/{(1 - 0.96)/(25 - 2 - 14)},

= 264.

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The given question is incomplete, the complete question is

A soft drink bottler is interested in predicting the amount of time required by the route driver to service the vending machines in an outlet. The industrial engineer responsible for the study has suggested that the two most important variables affecting the delivery time (Y) are the number of cases of product stocked (X₁) and the distance walked by the route driver (X₂).

The engineer has collected 25 observations on delivery time and multiple linear regression model was fitted Y' = 2.341 + 1.616×X₁ + 0.144×X₂. and R² = 96%

(a) Write down the model and then predict the delivery time when number of cases of product stocked = 10 and the distance walked by the route driver = 250.

(b) Find the adjusted R² and test for the overall model significance at 2.5% level.

The z-score for a backyard structure costing $2,300 is approximately -0.86.

To find the z-score for a backyard structure costing $2,300, we can use the formula:

z = (x - μ) / σ

where x is the value, μ is the mean, and σ is the standard deviation.

Given that the mean price for a backyard structure is $3,500 and the standard deviation is $1,400, we can substitute these values into the formula:

z = (2,300 - 3,500) / 1,400

Calculating this expression, we get:

z = -0.857

Rounded to two decimal places, the z-score for a backyard structure costing $2,300 is approximately -0.86.

The z-score measures the number of standard deviations a value is from the mean. It is used to standardize values and compare them to a standard normal distribution with a mean of 0 and a standard deviation of 1.

In this case, the z-score tells us how many standard deviations $2,300 is from the mean price of $3,500. A negative z-score indicates that the value is below the mean.

By calculating the z-score, we can determine the relative position of $2,300 compared to the distribution of backyard structure prices. A z-score of -0.86 suggests that $2,300 is approximately 0.86 standard deviations below the mean.

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Therefore, the correct option is O t≈ 6.2 seconds.

Given, T(t) = 16+ 44e-0.35t

We need to find how long will it take for the thermometer to reach 23°C. i.e,

we need to find t when T(t) = 23°C.

Substitute T(t) = 23°C in the given equation and solve for

t.23 = 16 + 44e-0.35t

t = -1/0.35

ln [(23-16)/44]≈ 6.2 seconds

Hence, the thermometer will take approximately 6.2 seconds to reach 23°C.

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The Florida Power and Light Company is testing solar panels from a company in Shanghai. They will reject the shipment if the defect rate is higher than 0.078. The decision to reject or accept depends on the hypothesis test results. If the defect rate is found to be higher, the shipment will be sent back. Otherwise, it will be accepted.


To determine the quality of solar panels received from a company in Shanghai, a hypothesis test is conducted with the null hypothesis H0: p = 0.078 (defect rate) and the alternative hypothesis Ha: p > 0.078 (defect rate). The significance level (α) is not mentioned, so we'll assume it to be 0.05 (5%).

If the null hypothesis H0 is rejected, it means there is strong evidence that the defect rate is higher than 0.078. In this case, the consequence would be that the entire shipment of solar panels would be sent back to the company in Shanghai, as the defect rate is higher than what the Florida Power and Light Company is willing to accept. This decision ensures that only panels with an acceptable defect rate are used.

If the null hypothesis H0 is not rejected, it means there is insufficient evidence to conclude that the defect rate is higher than 0.078. In this case, the consequence would be that the shipment of solar panels would be accepted by the Florida Power and Light Company. They would continue with the purchase, despite the defect rate, as it falls within their acceptable range.

To conduct the hypothesis test, a random sample of solar panels would be obtained from the shipment and carefully inspected. The defect rate in the sample would be calculated, and based on this information, a statistical test, such as a one-sample proportion test or a chi-square test, could be used to assess whether the defect rate is significantly different from 0.078.

The decision to reject or not reject the null hypothesis depends on the sample data and the chosen significance level. If the p-value (probability value) associated with the test statistic is less than the significance level, we would reject the null hypothesis. Otherwise, if the p-value is greater than or equal to the significance level, we would fail to reject the null hypothesis.

It's important to note that the consequences of the decision to reject or not reject the null hypothesis depend on the specific requirements and standards set by the Florida Power and Light Company. If they have strict quality control measures and are unwilling to accept defect rates higher than 0.078, rejecting the null hypothesis would be the appropriate decision.

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1. The total number of students who took the exam is 40 is option c.

2. The probability that the five cards picked are suited is approximately is option d. 0.0020.

Probability is a branch of mathematics that deals with the likelihood of events occurring. In this response, we will provide detailed solutions to two probability problems. We will explain the steps involved in solving each problem using mathematical terms.

Solution to Problem 1:

Let's denote the number of students who took the exam as 'x.' We are given that the probability of passing Chemistry is 70% and Physics is 50%. None of the students failed in both subjects, and 8 students passed both subjects.

To solve this problem, we can use the principle of inclusion-exclusion. The principle states that to find the total number of students who passed at least one subject, we need to sum the number of students who passed Chemistry, the number of students who passed Physics, and then subtract the number of students who passed both subjects.

Let's calculate the number of students who passed at least one subject:

Number of students who passed Chemistry = 0.7x

Number of students who passed Physics = 0.5x

Number of students who passed both subjects = 8

Total number of students who passed at least one subject = (Number of students who passed Chemistry) + (Number of students who passed Physics) - (Number of students who passed both subjects)

Substituting in the values, we have:

Total number of students who passed at least one subject = 0.7x + 0.5x - 8

Since none of the students failed in both subjects, the number of students who passed at least one subject is equal to the total number of students. Therefore, we can set the equation equal to 'x' and solve for it:

0.7x + 0.5x - 8 = x

Simplifying the equation:

1.2x - 8 = x

0.2x = 8

x = 8 / 0.2

x = 40

Therefore, the total number of students who took the exam is 40.

Hence, the answer to Problem 1 is option c. 40.

Solution to Problem 2:

We are given that we are picking five cards from a standard deck of 52 cards. We need to find the probability that all five cards picked are suited, meaning they all belong to the same suit.

To solve this problem, we can use the concept of combinations. The number of ways to choose five cards from a particular suit is denoted as C(13, 5), as there are 13 cards in each suit (hearts, diamonds, clubs, spades) and we need to choose 5 cards. Similarly, the total number of ways to choose any five cards from the deck is C(52, 5).

The probability of picking five suited cards can be calculated as:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Number of favorable outcomes = Number of ways to choose 5 cards from a single suit = C(13, 5)

Total number of possible outcomes = Number of ways to choose any 5 cards from the deck = C(52, 5)

Using the formula for combinations, we have:

C(n, r) = n! / (r!(n-r)!)

Substituting in the values, we get:

Number of favorable outcomes = C(13, 5) = 13! / (5!(13-5)!)

Total number of possible outcomes = C(52, 5) = 52! / (5!(52-5)!)

Calculating the values:

Number of favorable outcomes = 1,287

Total number of possible outcomes = 2,598,960

Now, we can calculate the probability:

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = 1,287 / 2,598,960

Simplifying the fraction:

Probability ≈ 0.000495

Therefore, the probability that the five cards picked are suited is approximately 0.000495.

Hence, the answer to Problem 2 is option d. 0.0020.

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In order to solve the given triangle, we will use the Sine law and Cosine law.  Let's start with Sine Law:

According to Sine Law, [tex]a/sin A = b/sin B \\= c/sin C[/tex]

where A, B, and C are the angles opposite to sides a, b, and c respectively. So, we have[tex]a/sin A = b/sin 54°\\ = 5/sin C     ........(i)[/tex]

Now, let's find A using Cosine Law Cosine law states that:

[tex]c² = a² + b² - 2ab cos C[/tex] On substituting the values, we get

[tex]5² = 8² + b² - 2×8×b cos 54° b² - 16b cos 54° + 39 \\= 0[/tex]

On solving the above quadratic equation, we get b = 6.02 (approx.)

[tex]C = 180° - A - BC \\= 180° - 66.96° - 54°C \\= 59.04°[/tex]

Hence, the required sides and angles are given as:

[tex]b = 6.02 (approx.)\\C = 59.04°A\\ = 66.96°[/tex]

[tex](f - g)(x) = f(x) - g(x)  \\= (3x² - 7) - (x - 6)(f - g)(3) \\= f(3) - g(3)  \\= (3*3² - 7) - (3 - 6)(f - g)(3) \\= 14[/tex]

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The cabin pressure is lower than the outside pressure, the air in the bag will expand.

To calculate the new volume of air in the bag, we can use the ideal gas law equation: PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature.

Since we are assuming the temperature is constant, we can rearrange this equation to solve for V.P1V1 = P2V2, where P1 is the initial pressure, V1 is the initial volume, P2 is the final pressure, and V2 is the final volume.

Substituting the given values:P1 = 105.5 kPa,V1 = 140.1 mL,P2 = 70.0 kPa,

V2  = 105.5 kPa x 140.1 mL

V2  = 70.0 kPa x V2

V2 = 221.14 mL

The new volume of air in the bag is 221.14 mL.

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The effectiveness of a maintenance program is affected most by the reliability of installed, replaceable items. This is because the reliability of installed replaceable items plays a key role in the effectiveness of a maintenance program.

The effectiveness of a maintenance program depends upon various factors like the availability of spare parts, skill level of the maintenance personnel, reliability of installed replaceable items, etc.

The four options provided in the question are Availability of spare parts, Designed ease of installation and removal of components, Reliability of installed replaceable items, and Skill level of the maintenance personnel.

Out of these options, the reliability of installed replaceable items has the most significant effect on the effectiveness of a maintenance program. This is because if the installed replaceable items are not reliable, then the maintenance program will not be able to function properly.

In such a case, the maintenance personnel will have to spend more time and effort to maintain the equipment, leading to increased downtime and decreased efficiency.

In contrast, if the installed replaceable items are reliable, then the maintenance program will be able to function smoothly.

The maintenance personnel will be able to quickly and easily replace the faulty components, minimizing downtime and increasing efficiency.

Therefore, the reliability of installed replaceable items is the most critical factor affecting the effectiveness of a maintenance program.

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a. sin y = -tan y * cos y = (-3/4) * (-7/8) = 21/32

b. sin (2x) = 2 * (-5/7) * (-2/7) = -5/7

c. cos (x-y) = (-2/7) * (-7/8) + (-5/7) * (21/32) = 21/28

a) Since both angles lie in quadrant IV, we know that both sin x and tan y are negative. We can use the Pythagorean identity, sin^2 x + cos^2 x = 1, to solve for cos x.

cos^2 x = 1 - sin^2 x = 1 - (-5/7)^2 = 1 - (25/49) = 24/49

cos x = sqrt(24/49) = -2/7

We can also use the identity tan^2 y = 1 - cos^2 y to solve for cos y.

cos^2 y = 1 - tan^2 y = 1 - (-3/4)^2 = 1 - 9/16 = 7/16

cos y = sqrt(7/16) = -7/8

sin y = -tan y * cos y = (-3/4) * (-7/8) = 21/32

b) Since x is in quadrant IV, we know that sin (2x) is negative. We can use the double angle formula, sin (2x) = 2 * sin x * cos x, to solve for sin (2x).

sin (2x) = 2 * (-5/7) * (-2/7) = -5/7

c) Since x and y are in quadrant IV, we know that cos (x-y) is positive. We can use the difference of angle formula, cos (x-y) = cos x * cos y + sin x * sin y, to solve for cos (x-y).

cos (x-y) = (-2/7) * (-7/8) + (-5/7) * (21/32) = 21/28

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Option A is correct, the most general antiderivative is [tex]\frac{7}{3}. e^{\:3x}\:dx+6e^-^x+C[/tex].

To find the antiderivative of the given function, we can apply the power rule for integration.

The power rule states that if f(x) is of the form f(x)=axⁿ.

where a is a constant and  n is a real number (except -1), then the antiderivative of f(x) with respect to x is given by [tex]\frac{a}{n+1}x^{n+1}+C[/tex].

Applying the power rule to each term in the given function, we have:

[tex]\int \:\left(7e^{3x}\:-6e^{-x}\:\right)dx=\int \:\:7e^{\:3x}\:dx-\int 6e^{\:-x}\:dx[/tex]

Integrating each term individually:

[tex]\:\int \:7e^{\:3x}\:dx=\:\frac{7}{3}e^{\:3x}+C_1[/tex]

[tex]\:\int \:6e^{\:-x}\:dx=\:-6e^{\:-x}+C_2\\[/tex]

Combining the results, we get:

[tex]\int \:\left(7e^{3x}\:-6e^{-x}\:\right)dx=\frac{7}{3}. e^{\:3x}\:dx+6e^-^x+C[/tex]

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1. The \( z \)-score for an IQ test score of 145, when the mean is 100 and the standard deviation is 15, is 3.

The \( z \)-score (or standard score) is a statistical measure that indicates how many standard deviations a data point is above or below the mean of a distribution. It is calculated using the formula:

\[ z = \frac{{X - \mu}}{{\sigma}} \]

where \( X \) is the value of the data point, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation of the distribution.

In this case, we have an IQ test score of 145, a mean of 100, and a standard deviation of 15. To find the \( z \)-score, we substitute these values into the formula:

\[ z = \frac{{145 - 100}}{{15}} = \frac{{45}}{{15}} = 3 \]

The \( z \)-score of 3 indicates that the IQ test score of 145 is three standard deviations above the mean. This means that the score is relatively high compared to the average performance on the IQ test.

By knowing the \( z \)-score, we can compare IQ test scores from different distributions and determine the relative position of a score within a distribution. A positive \( z \)-score indicates a score above the mean, while a negative \( z \)-score indicates a score below the mean. In this case, a \( z \)-score of 3 shows that the IQ test score of 145 is significantly above the mean.

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The graph is attached. One of the feature is the vertex being at (3, -3)

The graph of the equation y = (x - 3)² - 3 is a parabola.

The features are

The term (x - 3)² represents a quadratic term, indicating that the graph will be a parabola.

The vertex of the parabola is located at the point (3, -3)

since the equation is in the form (x - h)² + k,

where (h, k) represents the vertex coordinates.

The coefficient of the quadratic term, (x - 3)², is positive, indicating that the parabola opens upward.

The constant term, -3, shifts the parabola downward by three units.

The graph is attached

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It is true that m(A) = n(A) for all A ∈ σ(C). It is given that (X, A) is a measurable space and C is an arbitrary subset of A. And also given that m and n are two σ-finite measures on (X, A) such that m(A)

= n(A) for all A ∈ C.

We are to determine whether m(A)

= n(A) for all A ∈ σ(C) or not.σ(C) is the σ-algebra generated by C. It contains all the countable unions and countable intersections of the sets in C. We are to prove that m(A)

= n(A) for all A ∈ σ (C).If A ∈ C, then m(A)

= n(A) [Given]If A ∉ C, then A is the countable union of disjoint sets of C such that A

= B1 ∪ B2 ∪ .......∞

Then, m(A) = ∑m(Bn)and n(A) = ∑n(Bn)Therefore, m(A)

= n(A) since m(Bn) = n(Bn) for all n ∈ N. Now, if m and n are finite measures, then also we can show that m(A)

= n(A) for all A ∈ σ(C). We can prove this using Monotone class theorem since both m and n are finite.

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If ANFO has a cost of $1500 per tonne, the cost of explosives per tonne of rock is $3240.

To determine the blasthole spacing, we need to consider the burden and spacing relationship, as well as the desired powder factor. The burden (B) and spacing (S) are related by the equation B = 1.2S. We can calculate the burden using this equation.

Given that the hole diameter is 250 mm, the bench height is 15 m, and the stemming column length is 4 m, the effective depth for the explosive is the bench height minus the stemming column length and subdrill, which is 15 m - 4 m - 1.5 m = 9.5 m.

To calculate the blasthole spacing, we use the formula:

Spacing (S) = Burden (B) + Hole Diameter (D) = B + D.

From the given burden and spacing relationship (B = 1.2S), we substitute B in terms of S:

S = 1.2S + 250 mm.

Simplifying the equation, we have:

0.2S = 250 mm.

Converting the units, we get:

0.2S = 0.25 m.

Solving for S:

S = 0.25 m / 0.2 = 1.25 m.

Therefore, the blasthole spacing is 1.25 meters.

To calculate the cost of explosives per tonne of rock, we use the powder factor, which is given as 0.8 kg/m³. The density of the rock is 2.7 g/cc.

To convert the density of rock from g/cc to kg/m³:

Rock density = 2.7 g/cc = 2700 kg/m³.

The amount of explosive required per cubic meter of rock is:

Explosive amount = Powder factor * Rock density = 0.8 kg/m³ * 2700 kg/m³ = 2160 kg/1000 m³.

Since the explosive density is given as 0.9 g/cc, which is equivalent to 900 kg/m³, the cost of explosives per tonne of rock can be calculated as:

Cost of explosives = (Explosive amount / 1000) * Cost of ANFO per tonne = (2160 / 1000) * $1500 = $3240.

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The correct option is A. Moving away from the calculated radius and height along the constraint curve increases the surface area both in the short term and the long term

To determine the dimensions of the right circular cylinder with the minimum surface area, subject to a volume constraint of 162 in³, we can use Lagrange multipliers. Let's go through the steps:

First, we start with the formula for the surface area of a right circular cylinder, which is given by:

S = 2πr^2 + 2πrh = f(r, h)

The volume of the cylinder is given as V = πr^2h = 162 in³, which gives us the constraint equation:

g(r, h) = πr^2h - 162 = 0

To apply Lagrange's method, we define the Lagrangian function:

L(r, h, λ) = f(r, h) - λ[g(r, h) - 162]

Taking the partial derivatives of L(r, h, λ) with respect to r, h, and λ, and setting them equal to 0, we have:

∂L/∂r = 4πr + λ(2πh) = 0

∂L/∂h = 2πr + λ(πr^2) = 0

∂L/∂λ = πr^2h - 162 = 0

Solving these equations simultaneously, we find the values of r, h, and λ:

r = 3

h = 18/π

λ = -6/π

Therefore, the closed right circular cylinder with the smallest surface area, while having a volume of 162 in³, has a radius of 3 and a height of 18/π.

By performing the second partial derivative test, we can confirm that this critical point corresponds to an absolute minimum. Evaluating the determinant of the Hessian matrix:

(∂²L/∂r²)(∂²L/∂h²) - (∂²L/∂r∂h)² = 16π² > 0

Since the determinant is positive, it implies that the critical point indeed corresponds to a minimum.

Hence, the correct option is A. Moving away from the calculated radius and height along the constraint curve increases the surface area both in the short term and the long term.

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The required solution fax ax dx = (5/3)x³ - 7.5x + C.

Given: √6x²dx=0

To calculate: 6x dx. We have to differentiate and simplify the above equation.

√6x²dx=0

=> ∫6x²^(1/2) dx=0

Using the power rule of integration, we have

∫xn dx= xn+1/(n+1)

Hence, ∫6x²^(1/2) dx

= 6x^(2+1)/(2+1) + C

= 2x³+C

As ∫6x²^(1/2) dx=0,

2x³=-C

Now, 6x dx

= d(2x³)/dx= 6x²

Using x²0x = ¹27 dx, we have:

2x³ = 27

=> x³=27/2

=> x=3∛2

∴ 6x dx = 6* (3∛2)²

= 54∛2

Hence, 6x dx = 54∛2.

Given: S₁x²ax- 6

To calculate: fax ax dx.

We have,

S₁x²ax- 6= (2.5)

S₁x²ax- (2.5)(6/2)

Using the formula S₁x²ax = x³/3, we have:

S₁x²ax- 6

= (2.5) x³/3 - 7.5

= 5x³/6 - 7.5

∴ fax ax dx= ∫5x³/6 - 7.5 dx

Using the sum rule of integration, we get: fax ax dx

= 5 ∫x³/6 dx - ∫7.5 dx

Now, ∫x³/6 dx= 6(x³/6)/3 + C

= x³/3 + C

On integrating -7.5, we get, -7.5x.

Now, fax ax dx= 5(x³/3) - 7.5x + C

= (5/3)x³ - 7.5x + C

Thus, fax ax dx = (5/3)x³ - 7.5x + C.

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The dot product of vectors v and w is 0. The angle between v and w is 90 degrees. The vectors v and w are orthogonal.

To find the dot product of two vectors, v and w, we multiply the corresponding components of the vectors and then sum them up. In this case, v = 7i + 5j and w = 5i - 7j. To calculate the dot product vw, we multiply the corresponding components:

vw = (7i)(5i) + (5j)(-7j) = 35i^2 - 35j^2

Since i^2 = j^2 = -1, the dot product simplifies to:

vw = 35(-1) - 35(-1) = -35 - (-35) = 0

Therefore, the dot product of v and w is 0.

To find the angle between v and w, we can use the formula:

cosθ = (v · w) / (|v| |w|)

where θ is the angle between the vectors, v · w is the dot product of v and w, and |v| and |w| are the magnitudes of v and w, respectively.

Since the dot product vw is 0 and both v and w have nonzero magnitudes, the numerator of the formula is 0. Therefore, cosθ = 0, which implies that θ is 90 degrees.

Hence, the angle between v and w is 90 degrees, indicating that the vectors are orthogonal.

In summary, the dot product of v and w is 0, the angle between v and w is 90 degrees, and the vectors v and w are orthogonal.

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Ario will be at approximately 4.6 meters from the starting point when Miguel starts the race.

From the number line from 0 to 25, and Ario is standing 3 meters from the starting point. Miguel offers Ario a head start, and Miguel will start when the ratio of Ario's completed meters to Ario's remaining meters is 1:4.4.

Assume that Ario has covered x meters when Miguel starts the race. The remaining distance for Ario would be 25 - x meters.

According to the given ratio, set up the equation:

x / (25 - x) = 1 / 4.4

To solve for x, cross-multiply:

4.4x = 25 - x

Combining like terms:

5.4x = 25

Dividing both sides by 5.4:

x ≈ 4.6296

Rounding to the nearest tenth:

x ≈ 4.6

Therefore, Ario will be at approximately 4.6 meters from the starting point when Miguel starts the race.

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Given: y'=8x²-3xTo find: All possible functions with the given derivative.So we need to find the integral of the given function y', which will give us the possible functions. Let us solve for it using integration;

∫y' dx = ∫8x²-3x dx= 8 ∫x²dx - 3 ∫xdx

= 8 [ x³/3] - 3 [ x²/2 ] + C

= (8/3) x³ - (3/2) x² + C

where C is a constant of integration.

So, the possible functions with the given derivative y' are;A. (8/3) x³ - (3/2) x² + CB. (8/3) x³ - (3/2) x² - (2/3) x + CC. (8/3) x³ - (3/2) x² + (2/3) x + CD. (8/3) x³ - (3/2) x² - (2/3) x + C

The answer is, A. (8/3) x³ - (3/2) x² + C.

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Therefore, the range of g(x, y) is [-1, 1].

Let's evaluate g(1, −2):g(1, -2) = sin(6(1) + 8(-2)) = sin(-10) = -0.54402...

Now, let's find the range of g(x, y):

The function g(x, y) = sin(6x + 8y) has a range of [-1, 1] in interval notation.

The sin function has a maximum value of 1 and a minimum value of -1.

Since 6x + 8y can take on any real value, g(x, y) can take on any value between -1 and 1, inclusive.

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The equation given to us is: -6x² - 5y² = 3We need to find dy/dx.  Here, we can see that y is an implicit function of x. So we need to differentiate both sides of the given equation with respect to x by using the chain rule.

Let's do that: -6x² - 5y² = 3

Differentiating both sides

w.r.t x,

we get,-12x - 10y * dy/dx = 0

⇒ dy/dx = -12x / (10y)

Thus, we get the value of dy/dx which is

-12x/10y.

Therefore, the solution is:

dy/dx = -12x/10y,

where -6x² - 5y² = 3.

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A projectile fired at an angle of 30° to the horizontal with an initial speed of 560 m/s, Its range, R = 356963.0 m

The time of flight, T = 193.5 s The greatest height reached, H = 161213.7 m

For a particle with acceleration a(t) = -24k, initial velocity v(0) = i + j, and initial position r(0) = 0, The velocity of the particle, v(t) is -24tk + i + j.

The speed of the particle, v(t), is given as |v(t)| = |(-24tk + i + j)|

The position of the particle, r(t), is given as r(t) = -12t²k + tk + i + j For a football kicked at an angle of 30° to the horizontal with an initial speed of 61 ft/s.

The horizontal distance it travels while it is in the air is approximately 109.397 ft (to three decimal places).

The farthest from the goal post that the kick can originate and score a field goal is 17 ft (as a whole number).

For a particle that moves on the ellipse x² + 2 = 1 so that at time 1 ≥ 0, the position is given by the vector 2-1² 4t -i + 2+1² 2+12,

The velocity vector is given as v(t) = -4ti + 2j

The particle is not at rest  The coordinates of the point (x, y) that the particle approaches as increases without bound is (∞, 0).

For a projectile fired at an angle of 30° to the horizontal with an initial speed of 560 m/s, Its range, R = 356963.0 m

The time of flight, T = 193.5 s The greatest height reached, H = 161213.7 m

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A) F is conservative, the potential function is 8x² - 2xy + C₁. B) F is conservative, the potential function is 4y² + 4.5x² + C₂. C) F is conservative, the potential function is 4x² + 4.5y² + 0.5z² + C₃. D) F is conservative, the potential function is 16xsin(y) - 2y² + C₄. E) F is not conservative, no potential function.

A. F(x,y) = (16x - 2y)i + (-2x + 8y)j

To determine if F is conservative, we need to compute the curl of F:

curl F = (∂Fₓ/∂y - ∂Fᵧ/∂x)k

= (-2 - (-2))k

= 0

Since the curl of F is zero, F is conservative. To find a potential function f, we integrate the components of F with respect to their respective variables:

f(x,y) = ∫(16x - 2y) dx + C₁

= 8x² - 2xy + C₁

Therefore, the potential function for F(x,y) = (16x - 2y)i + (-2x + 8y)j is f(x,y) = 8x² - 2xy + C₁.

B. F(x,y) = 8yi + 9xj

To determine if F is conservative, we compute the curl of F:

curl F = (∂Fₓ/∂y - ∂Fᵧ/∂x)k

= (0 - 0)k

= 0

Since the curl of F is zero, F is conservative. To find a potential function f, we integrate the components of F with respect to their respective variables:

f(x,y) = ∫8y dy + ∫9x dx + C₂

= 4y² + 4.5x² + C₂

Therefore, the potential function for F(x,y) = 8yi + 9xj is f(x,y) = 4y² + 4.5x² + C₂.

C. F(x,y,z) = 8xi + 9yj + kz

To determine if F is conservative, we compute the curl of F:

curl F = (∂Fₓ/∂y - ∂Fᵧ/∂x)i + (∂Fₓ/∂z - ∂F_z/∂x)j + (∂Fᵧ/∂z - ∂F_z/∂y)k

= (0 - 0)i + (0 - 0)j + (0 - 0)k

= 0

Since the curl of F is zero, F is conservative. To find a potential function f, we integrate the components of F with respect to their respective variables:

f(x,y,z) = ∫8x dx + ∫9y dy + ∫z dz + C₃

= 4x² + 4.5y² + 0.5z² + C₃

Therefore, the potential function for F(x,y,z) = 8xi + 9yj + kz is f(x,y,z) = 4x² + 4.5y² + 0.5z² + C₃.

D. F(x,y) = (8sin(y))i + (-4y + 8xcos(y))j

To determine if F is conservative, we compute the curl of F:

curl F = (∂Fₓ/∂y - ∂Fᵧ/∂x)k

= (8cos(y) - 8cos(y))k

= 0

Since the curl of F is zero, F is conservative. To find a potential function f, we integrate the components of F with respect to their respective variables:

f(x,y) = ∫(8sin(y)) dx + ∫(-4y + 8xcos(y)) dy + C₄

= 8xsin(y) - 2y² + 8xsin(y) + C₄

= 16xsin(y) - 2y² + C₄

Therefore, the potential function for F(x,y) = (8sin(y))i + (-4y + 8xcos(y))j is f(x,y) = 16xsin(y) - 2y² + C₄.

E. F(x,y,z) = 8x²i - 2y²j + 4z²k

Since F is a function of x, y, and z, and the curl of F is not zero, F is not conservative. Therefore, there is no potential function f for F(x,y,z) = 8x²i - 2y²j + 4z²k. Hence, the answer is N.

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The general solution is

[tex]xₜ = 2^t/(1 - 1/2^t) (-2)^t + 2^t.yₜ = 2^t[/tex]

The given linear difference equations are

[tex]:x_{t+1}=-2x_t+y_ty_{t+1}=2y_t[/tex]

with initial values x_0=2 and y_0=1.

To solve the system of linear difference equations, we first solve for x_t in the first equation and substitute it into the second equation as follows:

[tex]xₜ₊₁ = -2xₜ + 2yₜ ; x₀ = 2 x₁ = -2(2) + 2(1) = -2x₂ = -2(-2) + 2(2) = 8x₃ = -2(8) + 2(2) = -12x₄ = -2(-12) + 2(2) = 28x₅ = -2(28) + 2(2) = -54[/tex]

Using the method of undetermined coefficients, we assume a solution of the form [tex]xt = C(-2)^t[/tex] and substitute into the difference equation

[tex](3)xₜ₊₁ = -2xₜ + 2yₜor C(-2)^(t+1) = -2C(-2)^t + 2yₜor C(-2) = -2C + 2yₜor yₜ = C(-2)/2 + C = -C + C(-2)^-1[/tex]

[tex]yₜ = 2^t[/tex]

Substituting, we get

[tex]2^t = -C + C(-2)^-1 or 2^t = C(1 - 1/2^t)or C = 2^t/(1 - 1/2^t)[/tex]

Substituting C gives the general solution

[tex]xt = 2^t/(1 - 1/2^t) (-2)^t + 2^t.[/tex]

The general solution is

[tex]xₜ = 2^t/(1 - 1/2^t) (-2)^t + 2^t.yₜ = 2^t[/tex]

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