The sum of two numbers is 35. The first number minus twice the second number is 8. Find the numbers.

If 2 similar "rectangular" shaped prisms have surface-area as 112 cm² and 1008 cm², then base perimeter of larger prism is 36 cm.

In order to calculate the Perimeter, we first calculate the "scale-factor" "k",

The "Scale-Factor" "k" is defined as the ratio of area of two figures which are similar,

So, (area of small rectangular prism)/(area of large rectangular prism) = k²,

Substituting the values of "Area",

We get,

⇒ k² = 112/1008,

⇒ k = 1/3,

Now, we use this "scale-factor" to find the value of the length and width of the "large-rectangular-prism".

For the length:

⇒ (length of small rectangular prism)/(length of large rectangular prism) = k,

⇒ 4/x = k,

⇒ 4/x = 1/3,

⇒ x = 12 cm.

For the width,

⇒ (width of small rectangular prism)/(width of large rectangular prism) = k,

⇒ 2/y = 1/3,

⇒ y = 6.

We know that the Perimeter(P) of base of Larger-Prism is calculated by the formula : 2l + 2w,

Substituting the values of Length(l) = 12 cm and width(w) = 6 cm,

We get,

⇒ Perimeter = 2×12 + 2×6 = 24 + 12 = 36 cm.

Therefore, the required Perimeter is 36 cm.

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The value of the given integral is 116.

To evaluate this integral, by finding an antiderivative of the function and then evaluating it at the limits of integration.

To apply this theorem to our integral, we first need to find an antiderivative of the integrand (the expression inside the integral sign). To do this, we need to use the power rule of integration, which states that the integral of tn is (1/(n+1))tn+1 + C, where C is the constant of integration.

Using this rule, we can find the antiderivative of the integrand as follows:

∫(4 - 2t + 3t²) dt = 4t - t²+ t³ + C

where C is the constant of integration.

[tex]\int\limits^5_1[/tex] (4 - 2t + 3t²) dt = [4t - t² + t³]5^1

= [(4(5) - (5)² + (5)³) - (4(1) - (1)² + (1)³)]

= [(20 - 25 + 125) - (4 - 1 + 1)]

= [120 - 4]

= 116

Therefore, the value of the given integral is 116.

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Answer:

(5x)(2x) = 202.5

10x^2 = 202.5

x^2 = 20.25, so x = 4.5

The length of AD is 2 inches.

Let x be the length of AD. By the Pythagorean theorem in triangle ABD, we have:

[tex]BD^2 + x^2 = AB^2[/tex]

Substituting AB = BC, we get:

[tex]BD^2 + x^2 = BC^2[/tex]

Using the fact that triangle ABC is isosceles, we can use the Pythagorean theorem in triangle ABC to find BC:

[tex]BC^2 = AC^2 - AB^2 = 8^2 - AB^2[/tex]

Substituting this expression for[tex]BC^2[/tex] in the previous equation, we get:

[tex]BD^2 + x^2 = 8^2 - AB^2[/tex]

Since BD is the perpendicular bisector of AC, we have AD = DC = (AC/2) = 4 inches. Therefore, we can write:

AB = AD + DB = 4 + DB

Substituting this expression for AB in the previous equation, we get:

[tex]BD^2 + x^2 = 8^2 - (4 + DB)^2[/tex]

Simplifying this equation, we get:

[tex]BD^2 + x^2 = 16 - 8DB - DB^2 + x^2[/tex]

Solving for DB, we get:

[tex]DB = (16 - BD^2 - x^2)/(8 + DB)[/tex]

Now, we can use the fact that BD is the perpendicular bisector of AC to write:

[tex]BD^2 = AD \times DC = 4x[/tex]

Substituting this expression for[tex]BD^2[/tex] in the previous equation, we get:

[tex]4x = (16 - 4x - x^2)/(8 + DB)[/tex]

Simplifying this equation, we get:

[tex]32x + 4x^2 = 16 - 4x - x^2[/tex]

Solving for x, we get:

x = 2 inches

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The total pounds of hamburger meat purchased by Sara is equal to 3 and 33/40 pounds in mixed number from.

Weight of one package of hamburger meat =2 and 3/8 pounds

Weight of second package of hamburger meat = 1 and 1/5 pounds

Weight of second package of hamburger meat = 1/4 pounds.

Converting the mixed numbers to improper fractions,

2 and 3/8 = 19/8

1 and 1/5 = 6/5

The total amount of hamburger meat that Sara purchased, add the weights of the three packages is equal to

2 and 3/8 + 1 and 1/5 + 1/4

= 19/8 + 6/5 + 1/4

To add these fractions and mixed numbers,

Convert them to a common denominator.

The least common multiple of 8, 5, and 4 is 40.

Now, rewrite the expression with the common denominator of 40,

= (19/8) × 5/5 + (6/5) × 8/8 + (1/4) × 10/10

= 95/40 + 48 / 40 + 10/40

= ( 95 + 48 + 10 ) / 40

=153/40

Simplifying this fraction to a mixed number, we get,

3 and 33/40 pounds

Therefore, Sara purchased a total of 3 and 33/40 pounds of hamburger meat.

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Answer:

X = -10

Step-by-step explanation:

work is in picture.

Hope this helps! Pls give brainliest!

We will use the future value of a series formula to solve this problem. The terms you want me to include are invests, investment, and compounded.

Here's a step-by-step explanation:

1. Bill invests $160 at the start of each month for 22 months. This is a regular investment, and we will treat it as an ordinary annuity.

2. The investment yields 0.5% per month, compounded monthly. We'll convert the percentage to a decimal by dividing it by 100, so the monthly interest rate (r) is 0.005.

3. We will use the future value of an ordinary annuity formula to find the value of the investment at the end of 22 months:

FV = P * [(1 + r)^n - 1] / r

Where FV is the future value of the investment, P is the monthly investment ($160), r is the monthly interest rate (0.005), and n is the number of months (22).

4. Plug in the values and calculate:

FV = 160 * [(1 + 0.005)^22 - 1] / 0.005

FV = 160 * [(1.005)^22 - 1] / 0.005

FV = 160 * [1.113688 - 1] / 0.005

FV = 160 * 0.113688 / 0.005

FV = 3,627.232

The value of Bill's investment at the end of 22 months is approximately $3,627.23.

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The third side of a triangle must be an integer, so the smallest possible integer solution is 5.

There are infinitely many possible third side lengths for a triangle with sides of length 5.3 and 0.4. To determine the third side length, we need to use the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side.

Thus, we have two inequalities:

0.4 + x > 5.3

5.3 + x > 0.4

Simplifying each inequality, we get:

x > 4.9

x > -4.9

The third side must be an integer, so the smallest possible integer solution is 5. Therefore, the length of the third side is 5 units.

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The point on the y-axis that lies on the line passing through point g and is parallel to line df is (0, 9).

To find the point on the y-axis that lies on the line passing through point g and is parallel to line df, we first need to determine the slope of line df. Since the line is parallel to the new line passing through point g, the slope will be the same.

Once we have the slope, we can use point-slope form to find the equation of the new line. Then, we can set x=0 (since we want to find the point on the y-axis) and solve for y to find the y-coordinate of the point.

So, let's begin.

First, let's find the slope of line df. We can use the formula:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are any two points on the line. We can use the points d and f, which are (5, 3) and (10, 8), respectively.

slope = (8 - 3) / (10 - 5) = 1

So the slope of line df is 1.

Now, using point-slope form, we can find the equation of the new line passing through point g (which is (-2, 7)) and having a slope of 1. The formula for point-slope form is:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is any point on the line (in this case, point g). Substituting in our values, we get:

y - 7 = 1(x - (-2))
y - 7 = x + 2
y = x + 9

So the equation of the new line is y = x + 9.

To find the point on the y-axis that lies on this line, we set x=0:

y = 0 + 9
y = 9

So the point on the y-axis that lies on the line passing through point g and is parallel to line df is (0, 9).

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The probability of return was the least for the Dining hall method and greatest for the Class delivery method. Therefore, E is the correct answer here.

There cannot be a positive or negative association between a categorical and a numerical variable. here the delivery method is a categorical variable.

We cannot determine what the total number for any of the delivery methods was as we are only given the conditional probability of whether they were returned or not.

We can clearly see that the probability of return was the least for the Dining hall method and greatest for the Class delivery method. Therefore, E is the correct answer here.

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The process of determining the effect of changing objective function coefficients, right-hand side values of constraints, and decision variable values on a linear program is known as sensitivity analysis.

Sensitivity analysis helps to understand how changes in the input parameters affect the optimal solution of a linear program. By analyzing the sensitivity of the solution to changes in the parameters, decision-makers can gain insight into the behavior of the model and make more informed decisions.

Sensitivity analysis involves computing the range of values over which the current optimal solution remains optimal, known as the range of optimality. It also involves computing the shadow prices, which indicate the change in the optimal objective function value per unit change in the right-hand side of a constraint. The shadow prices can help decision-makers understand the value of additional resources or the cost of resource shortages.

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Okay, here are the steps to find the probability that Josie will be placed in an A group with an even number:

1) There are 2 possible letters (A or B) and 6 possible numbers (1-6) for the groups. So there are 2 * 6 = 12 possible groups total.

2) Half of the groups (6 groups) will be A groups.

3) Half of the groups (6 groups) will have an even number (2, 4, 6)

4) So there are 6 * 6 / 12 = 1/2 = 0.5 probability that a randomly selected group will be an A group with an even number.

5) Therefore, the probability that Josie will be placed in an A group with an even number is 0.5.

Work shown:

Total groups: 12

A groups: 6

Even number groups: 6

Probability A group with even number: 6 * 6 / 12 = 1/2 = 0.5

Prob Josie in A even group: 0.5

0.5

Let me know if you have any other questions!

Answer: 1/6

Step-by-step explanation:

There are six possible outcomes when rolling a number cube, and two possible outcomes when drawing a letter tile.

To determine the probability that Josie will be placed in an A group with an even number, we need to find the number of outcomes that satisfy this condition and divide by the total number of possible outcomes.

The possible outcomes that satisfy the condition are:

The total number of possible outcomes is:

Therefore, the probability that Josie will be placed in an A group with an even number is:

Therefore, the probability that Josie will be placed in an A group with an even number is 1/6.

To find the amount of the 1.6 thousand gallons left at the end of the week, we need to determine the demand for the week and subtract it from the initial stock.

First, let's define the function h(x) as the amount left when the demand is x.

1. Identify the demand function for the week. This information is missing in your question, but let's assume it is given as d(x).

2. Calculate the demand for the week by evaluating the function d(x) at the end of the week. Let's assume the week is represented by the variable "t". Evaluate d(t) to find the demand for the week.

3. Subtract the demand for the week from the initial stock to find the remaining amount: h(t) = 1.6 thousand gallons - d(t).

4. Round your answer to three decimal places to get the final result.

Without the specific demand function, I cannot provide a numerical answer.

However, you can follow these steps with the given demand function to find the amount of the 1.6 thousand gallons expected to be left at the end of the week.

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The zeros of the polynomial f(x) are the values of x that make f(x) equal to zero. We can find the zeros of f(x) by setting the polynomial equal to zero and solving for x:

f(x) = (x-5)²6 (x²-25)²7 = 0

The polynomial f(x) has two factors, each of which contributes to the zeros of the polynomial:

Factor 1: (x-5)²6

This factor is equal to zero when x-5=0, or x=5. Therefore, the polynomial f(x) has a zero of multiplicity 6 at x=5.

Factor 2: (x²-25)²7

This factor is equal to zero when x²-25=0, or x=±5. Therefore, the polynomial f(x) has two more zeros at x=±5. Each of these zeros has a multiplicity of 7, since the factor (x²-25) is raised to the 7th power.

Therefore, the zeros of f(x) are x=5 and x=±5, with multiplicities of 6 and 7, respectively.In summary

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The circumcenter of a right triangle will lie on the hypotenuse of the triangle.

The circumcenter is the point at which the perpendicular bisectors of the sides of a triangle intersect. In a right triangle, the perpendicular bisectors of the legs intersect at the midpoint of the hypotenuse. This point is equidistant from all three vertices of the right triangle and is the circumcenter of the triangle.

Since the midpoint of the hypotenuse lies on the hypotenuse itself, the circumcenter of a right triangle must also lie on the hypotenuse. In fact, the circumcenter of a right triangle coincides with the midpoint of the hypotenuse. This is a unique feature of right triangles, and it can be used to find the circumcenter and other important properties of these triangles.

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The current utilization of the road is 0.5833 or 58.33%. To calculate the current utilization of the road, we need to use the formula:

Utilization = Arrival rate x Drive time

Since we are assuming Poisson arrival and exponential drive times, we can use the following formulas:

Arrival rate = λ = 35 cars per hour
Drive time = 1/μ = 1/60 hours (since the drive time is 1 minute)

Therefore,

Utilization = 35 cars per hour x (1/60 hours)
Utilization = 0.5833 or 58.33%

So the current utilization of the road in Mountainside Village is 58.33%.
Hi! As the transportation director of Mountainside Village, we can calculate the current utilization of the road using the given terms. The peak traffic rate is 35 cars per hour, and the drive time without traffic is 1 minute (or 1/60 hours).

Since we're assuming Poisson arrival and exponential drive times, we can calculate the utilization (ρ) using the formula:

ρ = λ / μ

Here, λ represents the arrival rate (35 cars/hour), and μ represents the service rate, which is the inverse of the average drive time (1/60 hours).

So, μ = 1 / (1/60) = 60 cars/hour

Now, we can calculate the utilization:

ρ = 35 cars/hour / 60 cars/hour = 0.5833 (rounded to 4 decimal places)

Thus, the current utilization of the road is 0.5833 or 58.33%.

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The 99% confidence interval for the true proportion of people who prefer x to y is approximately (0.4163, 0.5637).

To calculate the 99% confidence interval for the true proportion of people who prefer x to y, we'll use the following steps:

1. Find the sample proportion (p-hat): Divide the number of people who prefer x by the total number of people surveyed.
p-hat = 147/300 ≈ 0.49

2. Determine the sample size (n): In this case, the researcher surveyed 300 people, so n = 300.

3. Find the standard error (SE) of the proportion: SE = sqrt((p-hat * (1 - p-hat)) / n)
SE ≈ sqrt((0.49 * 0.51) / 300) ≈ 0.0286

4. Identify the 99% confidence level (z-value): For a 99% confidence interval, the z-value is 2.576 (from the standard normal distribution table).

5. Calculate the margin of error (ME): ME = z-value * SE
ME = 2.576 * 0.0286 ≈ 0.0737

6. Determine the lower and upper bounds of the 99% confidence interval:
Lower Bound = p-hat - ME ≈ 0.49 - 0.0737 ≈ 0.4163
Upper Bound = p-hat + ME ≈ 0.49 + 0.0737 ≈ 0.5637

So, the 99% confidence interval for the true proportion of people who prefer x to y is approximately (0.4163, 0.5637).

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The total surface area of the pyramid is 817 ft squared.

The surface area of the pyramid Jane is painting can be calculated as follows:

area of the pyramid = A + 1 / 2 ps

where

Therefore,

A = 19² = 361 ft²

p = 4(19) = 76 ft

s = 12 ft

Total surface area of the pyramid =  361 + 1 / 2 (76)(12)

Total surface area of the pyramid = 361 + 456

Total surface area of the pyramid = 817 ft²

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The correct answer is a: every individual has an equal chance of being selected. A random sample is a subset of a population that is selected in a way that each member of the population has an equal probability of being chosen.

The randomness of the sample helps to ensure that the results obtained from the sample are representative of the entire population.

Option b, the probabilities cannot change during a series of selections, is a requirement for independent events but not necessarily for a random sample.

Option c, there must be sampling with replacement, is not always required for a random sample. Sampling with replacement means that after an individual is selected, they are returned to the population and can be selected again. This is not always necessary for a random sample.

Therefore, the correct answer is a: every individual has an equal chance of being selected.

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The value of a is -5 after Fiona divided 3x² + 5x - 3 by 3x+2.

To find the value of a, we need to perform a polynomial division of 3x² + 5x - 3 by 3x + 2. The result of the division is a polynomial plus a remainder, which should be equal to ax + b for some constants a and b. The constant b represents the remainder over the divisor, so we can set the expression equal to b to find its value. When dividing 3x² + 5x - 3 by 3x + 2 using polynomial long division, we get:

            x                

3x + 2 | 3x² + 5x - 3

            - 3x² - 2x    

             3x - 3

            3x + 2        

                -5

Therefore, the remainder is -5, which is represented by the expression a = -5.

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The smallest irrational number from the given set of irrational numbers  is -√27

To determine the smallest irrational number from the given options, we need to identify which numbers are irrational. A rational number is any number that can be expressed as a ratio of two integers, while an irrational number cannot.

√5 is an irrational number because it cannot be expressed as a ratio of two integers. Similarly, -√27 is also irrational, since it cannot be expressed as a ratio of two integers.

√25, on the other hand, is a rational number because it equals 5 which can be expressed as a ratio of two integers, 5/1. 1/4 is also a rational number since it equals 0.25, which can be expressed as a ratio of two integers, 1/4. 0.2345678... appears to be a decimal expansion that goes on indefinitely, but it is rational since it can be expressed as a ratio of two integers by finding the pattern of the repeating digits.

Therefore, the smallest irrational number from the given options is -√27 since it is the only negative irrational number and has the smallest absolute value among the irrational numbers given.

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To find the percentage of commuters who take more than 45 minutes to get to school, we need to calculate the z-score first:

z = (45 - 32) / 11 = 1.18

Using a standard normal distribution table or calculator, we can find that the percentage of commuters who take more than 45 minutes to get to school is approximately 12.22%.

To find the time for the fastest 20% of all commuters to school, we need to calculate the z-score for the 20th percentile:

z = invNorm(0.2) = -0.84

The time for the fastest 20% of all commuters can be calculated using the formula:

x = μ + zσ = 32 + (-0.84) * 11 = 22.36 minutes

Therefore, the time for the fastest 20% of all commuters to school is approximately 22.36 minutes.

The 68-95-99.7% rule states that for a normally distributed data set, approximately 68% of the data falls within one standard deviation (σ) of the mean (μ), approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations of the mean.

Since we know that the mean time to commute to school is 32 minutes with a standard deviation of 11 minutes, we can apply the 68-95-99.7% rule to find the range of time for 95% of commuters:

Within one standard deviation (σ) of the mean (32 ± 11), approximately 68% of commuters take between 21 and 43 minutes to get to school.

Within two standard deviations of the mean (32 ± 2*11), approximately 95% of commuters take between 10 and 54 minutes to get to school.

Therefore, the statement "this shows us that 95% of the time to commute will be between 10 and 54 minutes to get to school" is correct.

To determine the longest 1% of the time to commute to school, we need to calculate the z-score for the 99th percentile:

z = invNorm(0.99) = 2.33

The longest 1% of the time to commute to school can be calculated using the formula:

x = μ + zσ = 32 + (2.33) * 11 = 57.63 minutes

Therefore, the longest 1% of the time to commute to school is approximately 57.63 minutes.

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On dividing the given two polynomials, we get another polynomial [tex]2x^{4} + 5x^{2}[/tex].

We are given two polynomials. One is 2[tex]x^{5}[/tex] + 5[tex]x^{3}[/tex] and the other polynomial is x. We have to divide the given two polynomials. As we know that when we add, subtract, multiply, or divide two polynomials, the answer is always a polynomial. Therefore, when we divide these two polynomials, the answer we get will also be a polynomial.

= [tex]\frac{2x^{5} + 5x^{3}}{x}[/tex]

We will take x common from the numerator so that we can cancel it out with the x present in the denominator. Therefore, taking x common;

[tex]\frac{x(2x^{4} + 5x^{2})}{x}[/tex]

Cancel out the term x from the numerator and denominator. By doing so, we get;

[tex]2x^{4} + 5x^{3}[/tex]

Therefore, on dividing the given two polynomials, we get the answer as a polynomial [tex]2x^{4} + 5x^{3}[/tex]

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The area of the parallelogram with vertices K(1,2,3), L(1,3,6), M(3,8,6), and N(3,7,3) is approximately 29.1547.

To find the area of the parallelogram, we need to find the length of one of its base vectors and the perpendicular height.

Let's first find one of the base vectors. We can take KL or MN as the base vector. Let's take KL.

The vector KL = L - K = (1, 3, 6) - (1, 2, 3) = (0, 1, 3).

Next, we need to find the perpendicular height of the parallelogram. We can do this by finding the cross-product of KL and KM, and then taking its magnitude.

The vector KM = M - K = (3, 8, 6) - (1, 2, 3) = (2, 6, 3).

Taking the cross product of KL and KM, we get:

KL x KM = |i j k|

|0 1 3|

|2 6 3|

= i(18) - j(6) + k(-2)

= (18, -6, -2)

The magnitude of KL x KM is:

[tex]|KL x KM| = √(18^2 + (-6)^2 + (-2)^2) = √(340) = 2*√(85)[/tex]

Therefore, the area of the parallelogram is:

Area = base x height = |KL| x |KL x KM| = [tex]√(0^2 + 1^2 + 3^2) * 2√(85) = 2√(10)√(85) = 2√(850) ≈ 29.1547[/tex]

So, the area of the parallelogram with vertices K(1,2,3), L(1,3,6), M(3,8,6), and N(3,7,3) is approximately 29.1547.

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Full Question: The Area of the Parallelogram with Vertices k(1,2,3), l(1,3,6), m(3,8,6), and n(3,7,3) is √265.

The original fraction is 5/2

A fraction can be defined as the part of a whole number, element or variable.

The different types of fractions are;

From the information given, we have that;

Let the numerator be x = 3x

x/3x - 2

. If 2 is added to the numerator and to the denominator

Then, we have;

x + 2/3x = 3/5

cross multiply the values, we get;

5(x + 2) = 3x(3)

expand the bracket

5x + 10 = 9x

collect the like terms

-4x = -10

Make 'x' the subject of formula

x = 5/2

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These prompts range from Similarity of triangles to similarity of angles, and transformations here are their answers.

6) ΔADE is similar to Δ ABC on the basis of proportionality.

AB/AC = AD/AE

4/8 = 2/4

1/2 = 1/2

hence on the bases on similar ratios, they are proportional.

7) ABCD transforms to EFGH on the bases of Option C.

8) the following statements about th parallel lines are true:
m∠2=m∠6

m∠1+m∠6 = 180°
x= 120°

If place the spape ABCD on the origin and rotate 180°, them move down ward the axis by 2 units to (0, -4) and two units to the right to (4, -4) you will get the result on the screen hence, the correct answer is Option C.

9) m∠1 = 35°, m∠2=30° (Option B)

Look closely, you would see that there is an angle on a straight line = 115°.

for the triangle of left it shares a co-linear angle with 115° lets call it x

So 80 + m∠1 + x = 180°
since x = 180-115 (angles on straight line) = 65°

Then

80 + m∠1 + 65 = 180

so m∠1= 180-80-65

m∠1 =35°

Using the same approach, we arrive at m∠2 =30°

Hence option B is the right answer.

10)  

(4x+7)  = (5x-10)  by virture of opposite angles.

Thus, (4x+7)  = (5x-10); after collecting like terms we have

⇒ 7+10 = 5x-4x
17= x

or x = 17
so: substituting x into the above expressions, we have

(4(17)+7) = 68 + 7 = 75°

75°

(5x-10) = 5(17) -10 = 85 -10 = 75°

Thus,  (4x+7)  = (5x-10) and are truly opposite angles.

Using x lets test to see if

(3x)° and (4x-14)° are alternate angle.

if they are, the should be equal.

Thus

3x = 3*17 = 51°
4x-14 = 4(17) -14 = 54°

Hence (3x)° ≠ (4x-14)°

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When testing a hypothesis about a population proportion with a sample size greater than 100, the proper test statistic to use is the z statistic. Option (a)

The z statistic is the appropriate test statistic to use when testing a hypothesis about a population proportion and the sample size is over 100. This is because the central limit theorem applies, which states that as the sample size increases, the sampling distribution of the sample proportion becomes approximately normal.

Therefore, the test statistic can be calculated as the difference between the sample proportion and the hypothesized population proportion, divided by the standard error of the sampling distribution. The z score can then be compared to the critical values or p-values from the standard normal distribution to determine the level of statistical significance and make conclusions about the hypothesis being tested.

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Full Question:  if You are testing a hypothesis about a population proportion and a sample size is over 100 what is the proper test statistic to use?

A) z statstic

B) chi-square

c) t statistic

d) not enough data

There were 2484 patrons in the theatre that night.

Let n represent the overall number of theatergoers that evening.

Let g represent the number of attendees in the gold seating, s represent the attendees in the silver seating, r represent the attendees in the red seating, and b represent the attendees in the black seating.

Consequently, n = g + s r b.

g = n because of the theatre goers are seated in the gold section.

r + b = n because of the customers are either in the red or the black seating.

The answer is obvious: b = 138.

As a result, r + b = n changes to

r + 138 = n, or r =  n - 138.

Since

n = n + 3(n - 138) + (n - 138) + 138

n = n + n - 414 + n - 138 + `138

n = n + n - 414

n = n - 414

n = 414

n = 2484

Therefore, there were 2484 patrons in the theatre that night.

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a. The velocity at time t is v(t) = t² + 2t - 15 m/s.

b. The distance travelled during the given time interval is [tex]\frac{25}{3}[/tex] meters (or approximately 8.33 meters).

(a) To find the velocity at time t, we need to integrate the acceleration function a(t) from 0 to t and add the initial velocity v(0):

[tex]v(t) = \int a(t) \, dt + v(0) = \int (2t + 2) \, dt - 15 = t^2 + 2t - 15[/tex]

So the velocity at time t is v(t) = t² + 2t - 15 m/s.

(b) To find the distance travelled during the given time interval, we need to integrate the velocity function v(t) from 0 to 5:

s(5) - s(0) = ∫v(t) dt from 0 to 5

Using the formula for v(t) from part (a), we have:

s(5) - s(0) = ∫(t² + 2t - 15) dt from 0 to 5

[tex]\int_{0}^{5} \left( \frac{t^3}{3} + t^2 - 15t \right) \, dt[/tex]

[tex]= \frac{25}{3}+25-75-\frac{0}{3}+0-0=\frac{25}{3}[/tex]

As a result, the distance covered in the allotted time is [tex]\frac{25}{3}[/tex], or roughly 8.33 metres.

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