A cone has a height of 15 feet and a diameter of 12 feet. What is its volume?

To test the claim that having lights on at night increases weight gain, we can conduct a two-sample t-test with unequal variances.

Let μ1 be the population mean weight change for mice living in bright light and μ2 be the population mean weight change for mice living in a normal light/dark cycle. The null hypothesis is H0: μ1 - μ2 = 0 (there is no difference in weight gain between the two groups) and the alternative hypothesis is Ha: μ1 - μ2 > 0 (mice in bright light gain more weight).

Using the given data, we can calculate the sample means and standard deviations:

x1 = 2.312 kg, s1 = 1.052 kg (for the sample of 10 mice in bright light)
x2 = 1.062 kg, s2 = 0.598 kg (for the sample of 8 mice in normal light/dark cycle)

We can then calculate the test statistic t:

t = (x1 - x2) / √(s1^2/n1 + s2^2/n2) = (2.312 - 1.062) / √(1.052^2/10 + 0.598^2/8) = 2.840

The degrees of freedom for the t-test is approximately given by the Welch-Satterthwaite equation:

df = (s1^2/n1 + s2^2/n2)^2 / (s1^4/(n1^2*(n1-1)) + s2^4/(n2^2*(n2-1))) = (1.052^2/10 + 0.598^2/8)^2 / (1.052^4/(10^2*9) + 0.598^4/(8^2*7)) = 14.867

Using a t-distribution table or calculator with df = 14.867 and a one-tailed test at α = 0.06 (equivalent to a critical t-value of 1.796), we find the p-value to be p = 0.006. Since this p-value is less than the significance level of 0.06, we reject the null hypothesis and conclude that there is evidence to support the claim that mice in bright light gain more weight than those in a normal light/dark cycle.

Note that the 6% level of significance is not a commonly used level and may be too liberal or too conservative depending on the context. It is important to consider the practical significance of the result and the potential for type I and type II errors.

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We have found that the remainder when x³ + 3x² + 3x + 1 is divided by x + 1 is 0.

The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x - a, the remainder of that division will be equivalent to f(a).

Given:

f(x) = x³ + 3x² + 3x + 1

We first calculate  the remainder of polynomial f(x) when divided by (x + 1).

We apply  the Remainder theorem, the remainder of f(x) when divided by (x - r) is f(r).

we then determine the value of f(-1).

Substituting value of  x = -1 is given polynomial f(x).

f(x) = x³ + 3x² + 3x + 1

f(-1) = (-1)³ + 3(-1)² + 3(*-1) +1

f(-1) = 0

In conclusion, the remainder when x³ + 3x² + 3x + 1 is divided by x + 1 is 0.

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Answer: When we use the change of variables u=x2,v=y3,w=z�=�2,�=�3,�=� to find the volume of the solid enclosed by the ellipsoid x24+y29+z2=1�24+�29+�2=1 above the xy−��− plane, we are essentially transforming the original equation of the ellipsoid into a new equation that is easier to work with.

Step-by-step explanation:

The new equation becomes u/4+v/9+w/1=1. We can now use this equation to find the volume of the solid by integrating over the region in uvw-space that corresponds to the region in xyz-space above the xy−��− plane. This region is a solid bounded by a plane, two planes perpendicular to the uvw-axes, and the surface of the ellipsoid. To integrate over this region, we can use triple integrals in uvw-space.
The triple integral would have limits of integration of 0 to 1 for u, 0 to (1-4u/9) for v, and 0 to sqrt(1-4u/9-v) for w. Integrating this triple integral would give us the volume of the solid enclosed by the ellipsoid above the xy−��− plane. In summary, the change of variables transforms the original equation into a simpler equation that can be used to set up a triple integral to find the volume of the solid. The region of integration in uvw-space corresponds to the region in xyz-space above the xy−��− plane, and we can use triple integrals to integrate over this region and find the volume.
Using the change of variables u = x^2, v = y^3, w = z, we can rewrite the equation for the ellipsoid as u/24 + v/29 + w^2 = 1. We want to find the volume of the solid enclosed by this ellipsoid above the xy-plane, which means we're looking for the region where w ≥ 0.

To do this, we will set up a triple integral over the given region using the Jacobian determinant to transform from (x, y, z) coordinates to (u, v, w) coordinates. The Jacobian determinant is given by:

J = |(∂(x,y,z)/∂(u,v,w))| = |(∂x/∂u, ∂x/∂v, ∂x/∂w; ∂y/∂u, ∂y/∂v, ∂y/∂w; ∂z/∂u, ∂z/∂v, ∂z/∂w)|

Computing the partial derivatives, we get J = |(1/2, 0, 0; 0, 1/3, 0; 0, 0, 1)| = 1/6.

Now, we can set up the triple integral:

Volume = ∫∫∫(u, v, w) dudvdw

The limits of integration for u will be 0 to 24, for v will be 0 to 29, and for w will be 0 to 1.

Volume = (1/6) ∫(0 to 24) ∫(0 to 29) ∫(0 to 1) dudvdw

Calculating this triple integral, we find the volume of the solid enclosed by the ellipsoid above the xy-plane:

Volume ≈ 58.8 cubic units.

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a) The critical value for a two-tailed test with a significance level of 0.01 and 11 degrees of freedom is 3.11 (found using a t-distribution table).

b) The test statistic for comparing two standard deviations is the F-statistic. The formula for calculating it is [tex]F = \frac{s1^2}{s2^2}[/tex], where s1 is the sample standard deviation of the first group (base product), s2 is the sample standard deviation of the second group (modified product), and the larger standard deviation is always in the numerator. Using the sample data given, we find:
s1 = 7 cm (from the base product)
s2 = 6.5 cm (from the modified product)
[tex]F = \frac{(7)^{2} }{(6.5)^{2} }  = 1.223[/tex]

c) To determine if there is a difference between the standard deviations, we compare the calculated F-statistic to the critical value we found in part a. Since our calculated F-value (1.223) is less than the critical value (3.11), we fail to reject the null hypothesis. Therefore, we conclude that there is not enough evidence to suggest that there is a significant difference between the standard deviations of the two products.

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The length of the lake of 4.042 miles in yards is 7113.92 yards

From the question, we have the following parameters that can be used in our computation:

Lake has a length of 4.042 mi.

This means that

Length = 4.042 miles

From the table of values:

To convert inches to feet, we multiply the length value by 1760

So, we have

Length = 4.042 * 1760 yards

Evaluate

Length = 7113.92 yards

Hence, the length is 7113.92 yards

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Answer:

(a) The total revenue from the sale of x thousand candy bars is equal to the product of the price charged for a candy bar and the number of candy bars sold. If p(x) is the price charged for a candy bar in cents, then the revenue R(x) in dollars is given by:

R(x) = (p(x) * 1000x) / 100

R(x) = (151 - x/10) * 100x

R(x) = 15100x - 1000x^2

Therefore, the expression for the total revenue from the sale of x thousand candy bars is R(x) = 15100x - 1000x^2 dollars.

(b) To find the value of x that leads to maximum revenue, we need to find the value of x for which R(x) is maximum. We can do this by finding the derivative of R(x) with respect to x, setting it equal to zero, and solving for x. So:

R'(x) = 15100 - 2000x

Setting R'(x) equal to zero, we get:

15100 - 2000x = 0

Solving for x, we get:

x = 7.55

Therefore, the value of x that leads to maximum revenue is 7.55 thousand candy bars.

(c) To find the maximum revenue, we substitute x = 7.55 into the expression for R(x):

R(7.55) = 15100(7.55) - 1000(7.55)^2

R(7.55) = $57042.50

Therefore, the maximum revenue is $57,042.50 when 7.55 thousand candy bars are sold.

Step-by-step explanation:

The maximum revenue from the sale of candy bars in the city is $84,375. a. The expression for total revenue from the sale of x thousand candy bars can be found by multiplying the price per candy bar by the number of candy bars sold:

Total revenue = p(x) * x

Substituting the given equation for p(x), we get:

Total revenue = (151 - x/10) * x

b. To find the value of x that leads to maximum revenue, we need to take the derivative of the revenue function and set it equal to zero:

d/dx (151x - x^2/10) = 0

Simplifying and solving for x, we get:

x = 750

c. To find the maximum revenue, we substitute the value of x obtained in part (b) into the revenue function:

Total revenue = (151 - 750/10) * 750 = $84,375

Therefore, the maximum revenue from the sale of candy bars in the city is $84,375.

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If a train travels 75 feet in 44 seconds. At the same speed, it travels 8.52 feet in 5 seconds

To find out how many feet the train will travel in 5 seconds at the same speed, first, we need to determine the speed of the train.

The train travels 75 feet in 44 seconds. To find the speed, we'll divide the distance traveled (75 feet) by the time taken (44 seconds):

Speed = Distance / Time
Speed = 75 feet / 44 seconds

Now, we can calculate the distance the train will travel in 5 seconds at the same speed:

Distance = Speed × Time
Distance = (75 feet / 44 seconds) × 5 seconds

The "seconds" unit cancels out, and we're left with:

Distance = (75 feet / 44) × 5

Now, we can calculate the distance:

Distance ≈ 8.52 feet

So, at the same speed, the train will travel approximately 8.52 feet in 5 seconds.

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The notation of number 25.67 x [tex]10^{-2[/tex] is not written in the scientific notation.

Scientific notation is a way of writing very large or very small numbers using powers of 10. It has the form [tex]a X 10^n[/tex], where a is a number between 1 and 10 (or sometimes between -1 and -10), and n is an integer.

As mentioned earlier, scientific notation has the form [tex]a X 10^n[/tex], where a is a number between 1 and 10 (or sometimes between -1 and -10), and n is an integer. But is should be reduced to one decimal number.

Thus, 25.67 x [tex]10^{-2[/tex] is not written in scientific notation.

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The possible values of the expression when the inequality is true, are:

(8, ∞).

Here we have the absolute value expression:

A = |x + 3|

And we know that x > 5, replacing that in the inequality, we will see that the lower bound of the possible values is:

A  >|5 + 3|

A > 8

Then the values allowed for the expression are all the values in the range (8, ∞).

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A teacher asked Dwayne to find the values of x and y in the triangles shown. The teacher provided the following information about the triangles. Triangle ABC is similar to triangle PQR. In triangle ABC, cos(C) = 0.92. Dwayne claims that the value of x can be determined.

Hence, the correct option is A.

Since triangles ABC and PQR are similar, their corresponding angles are congruent and their corresponding sides are proportional. Therefore, we can set up the following proportion we get

AB/BC = PQ/QR

We can also use the cosine law to relate the angle C in triangle ABC to the length of side AB and BC.

cos(C) = ([tex]AB^2 + BC^2 - AC^2[/tex])/(2AB*BC)

We are given that cos(C) = 0.92, and we know that AC = 20, AB = x, and BC = y, so we can substitute these values into the cosine law we get

0.92 = ([tex]x^2 + y^2[/tex] - 400)/(2xy)

Simplifying this equation, we get

([tex]x^2 + y^2[/tex] - 400) = 1.84xy

We can also use the given information to relate x and y we get

cos(R) = y/45

However, we cannot use this equation to solve for y because we do not know the value of cos(R).

Therefore, Dwayne is correct in claiming that we can determine the value of x using the cosine law, but we cannot determine the value of y with the information provided.  His claim is correct because cos(C) = x/20 and 0.92 can be substituted for cos(C), but the cosine of angle R is not given for triangle PQR.

Hence, the correct option is A.

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a) Samantha will have $20,359.68 in the account after the last $175 deposit is made in seven years.

b) Samantha makes 175-dollar deposits at the beginning of each month for seven years, she will have $21,372.77 in the account after the last deposit is made.

We can use the formula for the future value of an annuity with monthly compounding to solve this problem. The formula is:

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

Where:

FV is the future value of the annuity

P is the regular payment or deposit

r is the annual interest rate

n is the number of compounding periods per year (12 for monthly compounding)

t is the total number of years

a. If Samantha makes 175-dollar deposits at the end of each month for seven years, the total number of deposits she will make is:

7 years x 12 months/year = 84 deposits

The regular payment or deposit is P = $175, the annual interest rate is r = 5.4%, and the number of compounding periods per year is n = 12. The total number of years is t = 7.

Using the formula above, we can calculate the future value of the annuity:

FV = [tex]$175 *[/tex] [tex](((1 + 0.054/12)^(12*7) - 1) / (0.054/12))[/tex]

FV = [tex]$175 *[/tex] (((1.0045)[tex]^84 - 1[/tex]) / (0.0045))

FV =[tex]$175 * (116.2269)[/tex]

FV = $20,359.68

Therefore, Samantha will have $20,359.68 in the account after the last $175 deposit is made in seven years.

b. If Samantha makes 175-dollar deposits at the beginning of each month for seven years, we need to adjust the formula above to account for the timing of the deposits. One way to do this is to use the formula:

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

Where the additional factor (1 + r/n) accounts for the fact that the deposits are made at the beginning of each month.

Using this formula, we get:

FV = [tex]$175 * (((1 + 0.054/12)^(12*7)[/tex] - 1) / (0.054/12)) * (1 + 0.054/12)

FV = [tex]$175 * (((1.0045)^84 - 1)[/tex] / (0.0045)) * 1.0045

FV = $175 * (122.2837)

FV = $21,372.77

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The number of possible combinations of three coins from the bag of 16 coins with different dates can be calculated using the formula for combinations, which is nCr = n! / r!(n-r)!, where n is the total number of objects, r is the number of objects to be chosen, and ! denotes factorial (the product of all positive integers up to the given number).

In this case, we have n = 16 (the total number of coins in the bag) and r = 3 (the number of coins to be chosen for each combination). Using the formula for combinations, we can calculate the number of possible combinations as follows:

nCr = 16! / 3!(16-3)!
nCr = (16 x 15 x 14) / (3 x 2 x 1)
nCr = 560

Therefore, 560 possible combinations of three coins can be chosen from the bag of 16 coins with different dates. These combinations could represent different historical events, significant dates, or other symbolic meanings depending on the dates inscribed on the coins. The calculation of combinations is an important concept in combinatorics and probability theory, and it has many real-world applications in fields such as statistics, economics, and computer science.

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Using permutation, the soccer coach can choose one starter and one reserve player for the certain position in 56 different ways.

The coach wants to choose and order two players from a group of 8. This is a permutation problem since order matters.

To find the number of ways to choose one starter and one reserve player from 8 candidate players, you can use the following steps:

The formula for the number of permutations of n objects taken r at a time is nPr = n!/(n-r)!.

Using this formula, we can calculate the number of ways the coach can choose and order two players:

8P2 = 8!/(8-2)! = 8!/6! = 8x7 = 56

Therefore, there are 56 ways the coach can choose and order a starter and reserve player for the position from a group of 8 candidates.

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Answer: 132 Ounces.

Step-by-step explanation: 1 pound = 16 ounces. 8 1/4 or 8.25 x 16 = 132

Seth weigh 132 ounce when he was born.

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For example,Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

We have,

Seth weighed 8 pounds when he was 8 1/4 pounds born.

So, the weight in ounce

= 8 1/4 x 16

= 33/4 x 16

= 33 x 4

= 132 ounce

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The bracket makes an angle of approximately 51.48° degrees with the wall.

First, we can use the Pythagorean theorem to find the distance between the end of the bracket and the wall:

[tex]d = \sqrt{((3.2 ft)^2 - (2.5 ft)^2) }[/tex]≈ 1.99 ft

Now we can use the definition of the tangent function to find the angle x:

tan(x) = opposite / adjacent = 2.5 ft / 1.99 ft

Taking the arctangent of both sides, we get:

x = tan⁻¹(2.5 ft / 1.99 ft) ≈ 51.48°

Therefore, the bracket makes an angle of approximately 51.48° degrees with the wall.

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The product of two term 88 x 45  would be equal to 3960.

Since Multiplication is the mathematical operation that is used to determine the product of two or more numbers.

When an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.

Given that 88 x 45

We need to simply multiply the term;

88 x 45

= 3960

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The distance between the opposite corners of the windowpane would be 17 inches.

If the windowpane is divided diagonally, we see that the distance between the opposite corners can be be the hypotenuse of a right angle triangle.

This allows us to use the Pythagorean theorem to find that distance between opposite sides. The distance is:

d ² = 15 ² + 8 ²

d ² = 225 + 64

d ² = 289

d = √ 289

d = 17 inches

In conclusion, the distance between the opposite corners is 17 inches.

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The probability of a randomly selected hair salon rating having at most 2 stars is 0.44,

The probability of having more than 3 stars is 0.47.

We have,
(a) To find the probability that the randomly selected hair salon rating has at most 2 stars, we need to add the probabilities for 1-star and 2-star ratings.

Based on the provided probability distribution,

P(X=1) = 0.25 and P(X=2) = 0.19.

The probability of a rating having at most 2 stars.
P(X ≤ 2) = P(X=1) + P(X=2)

= 0.25 + 0.19

= 0.44

(b)

To find the probability that the randomly selected hair salon rating has more than 3 stars, we need to add the probabilities for 4-star and 5-star ratings.

Based on the provided probability distribution, P(X=4) = 0.21 and P(X=5) = 0.26.

The probability of a rating having more than 3 stars.
P(X > 3) = P(X=4) + P(X=5)

= 0.21 + 0.26

= 0.47

Thus,
The probability of a randomly selected hair salon rating having at most 2 stars is 0.44, and the probability of having more than 3 stars is 0.47.

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The value of the average rate of change is,

⇒ f ' (x) = 14

We have to given that;

The function is,

⇒ f (x) = 3x² - 4

Now, We can formulate;

The value of the average rate of change as;

⇒ f ' (x) = f (4) - f (2) / (4 - 2)

⇒ f ' (x) = (3 × 4² - 4) - (3 × 2² - 4) / 2

⇒ f ' (x) = 44 - 16 / 2

⇒ f ' (x) = 28/2

⇒ f ' (x) = 14

Thus.,  The value of the average rate of change is,

⇒ f ' (x) = 14

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The minimum sample size needed to be 99% confident that the sample mean is within 4 hours of the true population mean is 27.

To calculate the minimum sample size needed to be 99% confident that the sample mean is within 4 hours of the true population mean with a population standard deviation of 8 hours, follow these steps:

1. Identify the desired confidence level: In this case, it is 99%.
2. Find the corresponding Z-score for the confidence level: For a 99% confidence level, the Z-score is approximately 2.576.
3. Identify the population standard deviation: In this case, it is 8 hours.
4. Identify the margin of error: In this case, it is 4 hours.
5. Use the following formula to calculate the sample size:
  Sample size (n) = (Z-score^2 * population standard deviation^2) / margin of error^2

Plugging in the values, we get:
n = (2.576^2 * 8^2) / 4^2
n = (6.635776 * 64) / 16
n = 26.543104

Since we need to round up to the nearest integer, the minimum sample size needed is 27.

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Answer:

The surface area of the shape is 1. 96   2. square inches.

1. 96

2. square

Step-by-step explanation:

We can see that the given right triangular prism is composed of the following regular 2D shapes whose areas add up to the total surface area of the prism

Total surface area of triangular prism = 12 + 16 + 20 + 48
= 96 square inches

Answer:

96 square inches

Step-by-step explanation:

To figure out how much surface area a right triangular prism has, you gotta break it down into a few 2D shapes. There's a tall rectangle that's 6 inches wide and 2 inches tall, which is 12 square inches. Then there's a wide rectangle that's 8 inches wide and 2 inches tall, which is 16 square inches. There's also a rectangle on the diagonal part of the prism that's 2 inches wide and 10 inches tall, which is 20 square inches. And don't forget the two triangles on the sides! Each triangle is 6 inches tall and 8 inches wide, which is 24 square inches each, for a total of 48 square inches. Add all those areas up and bam, you've got the total surface area of the triangular prism - which in this case is 96 square inches.

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The class limits for the class with the highest frequency is 65 up to 75. The correct answer is A.

The frequency distribution given in the question represents the number of textbooks and their corresponding costs. The distribution is divided into several classes, each representing a range of costs. The frequency for each class indicates how many textbooks fall within that range of costs.

The question asks us to find the class limits for the class with the highest frequency. We can see from the distribution that the class with the highest frequency is "65 up to 75", which has a frequency of 20.

The class limits for a given class are the lowest and highest values included in that class. In this case, the lower limit of the class "65 up to 75" is 65 (because it is the lowest value in that range), and the upper limit of the class is 75 (because it is the highest value in that range).

Therefore, the class limits for the class with the highest frequency are 65 (the lower limit) and 75 (the upper limit), and the correct answer is "65 up to 75".  The correct answer is A.

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p^2 + 4p - 48 = 0

Let's first express Aiman's age in terms of his younger brother's age. If Aiman is four years older than his younger brother, then we can write:

Aiman's age = younger brother's age + 4

Let p be the age of Aiman's younger brother. Then Aiman's age is p + 4.

The product of Aiman and his younger brother's ages is equal to their father's age, which is given as 48. So we can write:

(p + 4) * p = 48

Expanding the left-hand side:

p^2 + 4p = 48

Subtracting 48 from both sides:

p^2 + 4p - 48 = 0

This is a quadratic equation in terms of p.

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Answer:

Deon drove 270 miles.

Let x be the number of miles driven.

The total cost is 20.95 + 0.74x = 221.49

Subtracting 20.95 from both sides gives 0.74x = 200.54

Dividing both sides by 0.74 gives x = 270 miles

So the answer is 270

Answer:

271 miles

Step-by-step explanation:

The equation to find the total cost is

C = 20.95 + .74 m  where m is the number of miles

221.49 = 20.95 +.74m

Subtract 20.95 from each side

221.49 -20.95 = 20.95-20.95 +.74m

200.54 = .74m

Divide each side by .74

200.74/.74 = m

271 = m

The probabilities are:

(a) P(X = 0) ≈ 0.3139

(b) P(X ≥ 1) ≈ 0.6861

(c) P(X ≤ 1) ≈ 0.9862

We can model this situation using the hypergeometric distribution.

Let's define:

N = total number of manufacturing plants = 30

D = number of plants outside the country = 7

n = number of plants in the performance evaluation = 4

(a) Probability of including no plants outside the country:

We want to find P(X = 0), where X is the number of plants outside the country in the performance evaluation. This can be calculated using the hypergeometric distribution formula:

P(X = 0) = (C(23, 4) * C(7, 0)) / C(30, 4) = (23 choose 4) / (30 choose 4) ≈ 0.3139

(b) Probability of including at least 1 plant outside the country:

We want to find P(X ≥ 1). We can use the complement rule and find the probability of including no plants outside the country and subtract it from 1:

P(X ≥ 1) = 1 - P(X = 0) = 1 - (C(23, 4) * C(7, 0)) / C(30, 4) ≈ 0.6861

(c) Probability of including no more than 1 plant outside the country:

We want to find P(X ≤ 1). This can be calculated as the sum of P(X = 0) and P(X = 1):

P(X ≤ 1) = P(X = 0) + P(X = 1) = (C(23, 4) * C(7, 0)) / C(30, 4) + (C(23, 3) * C(7, 1)) / C(30, 4) ≈ 0.9862

Therefore, the probabilities are:

(a) P(X = 0) ≈ 0.3139

(b) P(X ≥ 1) ≈ 0.6861

(c) P(X ≤ 1) ≈ 0.9862

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Answer:

- 21

Step-by-step explanation:

let the integer be n , then

[tex]\frac{n}{7}[/tex] = - 3 ( multiply both sides by 7 to clear the fraction )

n = 7 × - 3 = - 21

that is the integer is - 21

The distance between the yacht and the kittiwake. is 160 m

The horizontal distance between Sharon and the yacht

tan 19 = 90 / distance between Sharon and the yacht

distance between Sharon and the yacht = 90 / tan 19

distance between Sharon and the yacht = 261.38 m

The horizontal distance between the Sharon and the kittiwake

tan 15 = distance between the Sharon and the kittiwake / 261.38

distance between the Sharon and the kittiwake = 261.38 x tan 15

distance between the Sharon and the kittiwake = 70.04 m

distance between the yacht and the kittiwake

= 90 + 70.04

= 160.04

= 160 m

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The probability that the distance from P to C is smaller than the square of the distance from P to D is 1/3.

Given a line segment CD of length 6.

A point P is chosen at random on CD.

Let C(0, 0) and D (6, 0).

Any point in between C and D will be of the form (x, 0).

So let P (x, 0).

Then using distance formula,

CP = √x² = x

PD = √(6 - x)² = 6 - x

CP < (PD)²

x < (6 - x)²

x < 36 - 12x + x²

x² - 13x + 36 > 0

(x - 9)(x - 4) > 0

x - 9 > 0 and x - 4 > 0

x > 9 and x > 4  

x > 9 is not possible.

Hence x > 4.

Possible lengths are 5 and 6.

Probability = 2/6 = 1/3

Hence the required probability is 1/3.

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The sandbox's 3/4ths fraction is full, the volume of sand can be found by multiplying the volume of the entire sandbox (0.33 ft³) by 3/4, which gives us 0.2475 ft³ of sand.

The area given (64 cm²) is the base of the sandbox, we can find the height of the sandbox in cm using the formula for the area of a rectangle: A = l × w.

Since the area is 64 cm², and we know that the sandbox has a rectangular base, we can assume the length and width are equal and each measure 8 cm.

Next, we need to convert the height of the sandbox from cm to feet. One foot is equal to 30.48 cm, so the sandbox is 0.33 feet tall (approximately). Since the sandbox is 3/4ths full, the volume of sand can be found by multiplying the volume of the entire sandbox (0.33 ft³) by 3/4, which gives us 0.2475 ft³ of sand.

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The sandbox was a foot tall, but the sand only was only filled up 3/4ths of the way with an area of 64 cm². How many cubic feet of sand are there in the box?

The final value is  0.67

(a) For a perpendicular rectangle without a common edge, the shape factor can be calculated using the formula:

f12 = min(w1, w2) / (h1 + h2)

where w1 and w2 are the widths of the rectangles and h1 and h2 are the heights.

In this case, the minimum width is 2 units, and the sum of the heights is 5 + 3 = 8 units. Therefore,

f12 = 2 / 8 = 0.25

(b) For parallel rectangles of unequal areas, the shape factor can be calculated using the formula:

f12 = (A2 / A1) * (h1 / h2)

where A1 and A2 are the areas of the rectangles and h1 and h2 are the heights.

In this case, the area of rectangle 1 is 24 square units (4 x 6) and the area of rectangle 2 is 16 square units (4 x 4). The heights are the same at 4 units. Therefore,

f12 = (16 / 24) * (4 / 4) = 0.67

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