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The hypothesis that the variance in the number of patients seen per day is equal to 10 cannot be rejected  based on the given data and using a level of significance of 0.05.

To determine if the variance in the number of patients seen per day is equal to a specific value, we can conduct a hypothesis test. Let's assume the null hypothesis is that the variance is equal to the specified value, and the alternative hypothesis is that the variance is not equal to the specified value.

We can use a chi-square test to test this hypothesis, where the test statistic is calculated as (n-1)*s²/σ², where n is the sample size, s² is the sample variance, and σ² is the hypothesized population variance. This test statistic follows a chi-square distribution with n-1 degrees of freedom.

Using a level of significance of 0.05, with 9 degrees of freedom (since there were 10 observations), the critical value for the chi-square distribution is 16.92.

Calculating the sample variance from the given data, we get s^2 = 4.44. Assuming the hypothesized population variance is 10, the test statistic is (9)*4.44/10 = 4.00.

Since the test statistic (4.00) is less than the critical value (16.92), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that the variance in the number of patients seen per day is not equal to 10.

In conclusion, based on the given data and using a level of significance of 0.05, we cannot reject the hypothesis that the variance in the number of patients seen per day is equal to 10.

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The surface area of the triangular prism is 75 ft squared.

The prism is a triangular base prism. The surface area of the prism can be found as follows:

surface area of the prism = (a + b + c)l + bh

where

Therefore,

a = 2 ft

b = 1.5 ft

c = 2.5 ft

l = 12 ft

Hence,

Surface area of the triangular prism = (2 + 1.5 + 2.5)12 + 2(1.5)

Surface area of the triangular prism = 6(12) + 3

Surface area of the triangular prism = 72 + 3

Surface area of the triangular prism = 75 ft²

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60 ounces of iodine worth 20 cents per ounce must be mixed with 40 ounces of iodine worth 15 cents per ounce to create a mixture worth 18 cents per ounce.

We have,

Let x be the number of ounces of iodine worth 20 cents per ounce that must be mixed.

The total amount of iodine after mixing is x + 40 ounces, and the total value of the mixture is (20x + 15(40)) cents.

The problem can be expressed as the equation:

(20x + 15(40))/(x + 40) = 18

Multiplying both sides by (x + 40) gives:

20x + 600 = 18(x + 40)

Expanding the right side gives:

20x + 600 = 18x + 720

Subtracting 18x and 600 from both sides gives:

2x = 120

Dividing both sides by 2 gives:

x = 60

Therefore,

60 ounces of iodine worth 20 cents per ounce must be mixed with 40 ounces of iodine worth 15 cents per ounce to create a mixture worth 18 cents per ounce.

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The solution of the given inequality is x less than 3/23.

The given inequality is 8>23x+5.

Subtract 5 on both the sides of an inequality, we get

8-5>23x+5-5

3>23x

Divide 23 on both the sides of an inequality, we get

3/23>x

x<3/23

Therefore, the solution of the given inequality is x less than 3/23.

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The sample data to estimate the population parameter with a certain degree of confidence.

Hypothesis testing is a statistical inference technique used to determine if a hypothesis about a population parameter is supported by the sample data. The hypothesis testing procedure consists of several steps:

State the null hypothesis and the alternative hypothesis: The null hypothesis (H0) is the hypothesis that there is no significant difference between the sample data and the population parameter. The alternative hypothesis (Ha) is the hypothesis that there is a significant difference between the sample data and the population parameter.

Choose a significance level: The significance level (α) is the probability of rejecting the null hypothesis when it is actually true. A commonly used significance level is 0.05.

Collect the sample data: The sample data should be collected randomly and should be representative of the population.

Calculate the test statistic: The test statistic is a numerical value calculated from the sample data that measures how well the sample data support the null hypothesis.

Determine the p-value: The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one calculated from the sample data, assuming the null hypothesis is true.

Make a decision: If the p-value is less than the significance level, reject the null hypothesis and accept the alternative hypothesis. If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.

To give an example, let's say we want to determine if the mean weight of apples produced by a particular orchard is different from 150 grams. We collect two random samples of apples, each with a sample size of 50. We set up the hypothesis as follows:

H0: μ = 150

Ha: μ ≠ 150

We choose a significance level of 0.05. We calculate the test statistic as follows:

t = (x - μ) / (s / sqrt(n))

where x is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Let's say we get a t-value of 2.5 and a p-value of 0.015. Since the p-value is less than the significance level, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean weight of apples produced by the orchard is different from 150 grams. We can then use the sample data to estimate the population parameter with a certain degree of confidence.

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The area of the donut ta that is left is 16.485 inches².

We have,

Donuts:

Diameter = 5 inches

Radius = 5/2 inches

The area of the donuts.

= πr²

= 3.14 x 5/2 x 5/2

= 19.625 inches²

Now,

The diameter of the hole in the donut = 2 inches

Radius = 2/2 = 1 inch

Area of the donut hole.

= 3.14 x 1 x 1

= 3.14 inches²

Now,

The area of the donut ta that is left.

= 19.625 - 3.14

= 16.485 inches²

Thus,

The area of the donut ta that is left is 16.485 inches².

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Therefore, the rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is $4500 is approximately 12.444 dollars per unit of revenue per dollar increase in cost.

The rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is $4500, we need to find dR/dC when C changes by 12 and R is 4500.

From the given equation, we know that:

C = [tex]R^2/112,000 + 5807[/tex]

Taking the derivative of both sides with respect to R, we get:

dC/dR = 2R/112,000

Solving for dR/dC, we get:

dR/dC = 1/(dC/dR) = 112,000/(2R)

When the cost per unit changes by $12, the new cost is:

C + dC = [tex]R^2/112,000 + 5807 + 12[/tex]

And when the revenue is $4500, we have:

R = 4500

Values into the expression for dR/dC, we get:

dR/dC = 112,000/(2 * 4500)

= 12.444

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The student who cut his time by the greatest percentage is

Student B.

We have,

Students A's time decreased by 0.125

= 0.125 x 100

= 12.5%

Student B's decreased by 1/6 which in decimal is

= 1/6

= 0.16667

= 16.66667%

and, Students C's time decreased by 15%.

= 15/100

= 0.15

Thus, the student who cut his time by the greatest percentage is

Student B.

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The solution is:

A) Deceleration of the car is -6.6667 m/s² while it came to stop.

B) The total distance the car travels is 200 meter during the 10 s period.

Here, we have,

Explanation:

Given Data

Initial velocity of the car () =   20.0 m/s

Final velocity of the car () = 0 m/s

Time (in motion) =7.00 s

Time (in rest) =3 s

To find - A) car's deceleration while it came to a stop

             B) the total distance the car travels in 10 s

A) The formula to find the deceleration is

Deceleration = (( final velocity - initial velocity ) ÷ Time)        (m/s²)

Deceleration = (() - ()) ÷ time     (m/s²)

Deceleration =  ( 0.0 - 20 ) ÷ 3      (m/s²)

Deceleration =   (- 20) ÷ 3  (m/s²)

Deceleration   =  - 6.6667 m/s²

(NOTE : Deceleration is the opposite of acceleration so the final result must have the negative sign)

The car's deceleration is  - 6.6667 m/s² while it came to a stop

B) The formula to find the distance traveled by the car is  

Distance traveled by the car is equals to the product of the speed and time

Distance = Speed × Time  (meter)

Distance = 20.0 × 10

Distance = 200 meters

The total distance the car travels during the period of 10 s is 200 meters.

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complete question:

A car is traveling at a 20.0 m/s for 7.00 s and then suddenly comes to a stop over a 3 s period.

a. What was the car’s deceleration while it came to a stop?

b. What is the total distance the car travels during the 10 s period?

Answer:

13.65

Step-by-step explanation:

25% + 15% = 35%

35% of 21 = 7.35

21 - 7.35 = 13.65

hope this helps :)

The area of the circle with a circumference of 6pi m is 9π square meters.

A circle is simply a closed 2-dimensional curved shape with no corners or edges.

The area of a circle is expressed mathematically as;

A = πr²

The circumference of a circle is expressed as:

C = 2πr

Where r is radius and π is constant pi.

We are given that the circumference of the circle is 6π m, so we can set up the equation and solve for the radius.

C = 2πr

6π = 2πr

Simplifying, we get:

r = 6π/2π

r = 3

Now that we know the radius of the circle, we can use the formula for the area of a circle:

A = πr²

Substituting the value of r, we get:

A = π(3)²

Simplifying, we get:

A = 9π

Therefore, the area is 9π m.

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We can conclude that for all integers a and b, if a|b, then a ≤ b.

To prove that for all m and n, if m ≠ n, then m+n ≠ Q, we will use proof by contradiction.

Assume that for some m and n, m ≠ n, and m+n = Q, where Q is a rational number. By the definition of a rational number, Q can be expressed as the ratio of two integers, p and q, where q ≠ 0.

Thus, we have:

m + n = p/q

Multiplying both sides by q, we get:

mq + nq = p

Rearranging, we get:

mq = p - nq

Since p, n, and q are integers, p - nq is also an integer. Therefore, mq is an integer.

But we know that m and n are integers and m ≠ n, which implies that m and n have different prime factorizations. Therefore, mq cannot be an integer, as it would require m and q to have a common factor, which is not possible.

This contradicts our assumption that m+n = Q, and hence, we can conclude that for all m and n, if m ≠ n, then m+n ≠ Q.

To prove that for all integers a and b, if a|b, then a ≤ b, we will use direct proof.

Assume that a and b are integers such that a|b, i.e., there exists an integer k such that b = ak.

To prove that a ≤ b, we need to show that a is less than or equal to k times a, i.e., a ≤ ka.

Dividing both sides of the equation b = ak by a (which is possible as a ≠ 0 since it is a divisor of b), we get:

b/a = k

Since k is an integer, we know that b/a is also an integer. Therefore, a must be less than or equal to b/a.

Multiplying both sides of the inequality a ≤ b/a by a (which is a positive number since a > 0), we get:

[tex]a^2 ≤ ab[/tex]

Since a and b are both positive integers, we know that [tex]a^2 ≤[/tex] ab implies that [tex]a ≤ b[/tex].

Therefore, we can conclude that for all integers a and b, if a|b, then a ≤ b.

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The exact area of the figure is 85.54 m².

The coordinates endpoints of the semicircle are (-3,-4) and (6,2).

∴ Length of the diameter of the circle = distance between the end points of diameter = √[6 -(-3))²+(-4-2)²] = 10.81 m

⇒ The radius of the semicircle = 10.81/2 = 5.45 m

∴ The area of the semicircle = πr²/2 = 3.14 × (5.45)² /2= 46.63 m²

Now one side of the rectangle = diameter of the semicircle

                                                   = 10.81 m

The endpoints of another side of the rectangle are (-3,-4) and (-1,-7)

∴  The length of this side = √[-3 -(-1))²+(-4-(-7)²] = 3.6 m.

∴ The area of the rectangle = product of sides = 3.6 × 10.81 = 38.91 m²

The total area of the figure = area of the semicircle + area of the rectangle

                                       = 46.63 + 38.91 =85.54 m²

Hence, the exact area of the figure is 85.54 m².

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Step-by-step explanation:

See image:

The fundamental period of [n] is N=72

(a) z(t) = 0 is a constant signal, which means it does not vary with time. A constant signal is not periodic because it does not repeat over time. Therefore, z(t) = 0 is not periodic.

(b) [n] = 1 sin[n] + 4 cos[-] 72- is a discrete-time signal, which means it is defined only at integer values of n. To determine whether it is periodic, we need to check whether there exists a positive integer N such that [n] = [n+N] for all integer values of n.

Using trigonometric identities, we can simplify [n] as follows:

[n] = 1 sin[n] + 4 cos[-] 72-
   = 2 sin[36-] cos[-] 36- + 2 cos[36-] sin[n]

Next, we can rewrite [n+N] using the same trigonometric identities:

[n+N] = 2 sin[36-] cos[-] 36- + 2 cos[36-] sin[n+N]

For [n] to be periodic with period N, [n] must be equal to [n+N] for all integer values of n. This means that the two expressions above must be equal for all n, which in turn means that sin[n] must be equal to sin[n+N] and cos[n] must be equal to cos[n+N] for all n.

Since sin and cos are periodic with period 2π, this condition is satisfied if and only if N is a multiple of 72, which is the least common multiple of 36 and 72. Therefore, the fundamental period of [n] is N=72.

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The value of probability that a customer wins a prize is,

⇒ 12.5%

We have to given that;

A jewelry store has each customer spin a spinner with 8 equal sections numbered 1, 1, 1, 2, 2, 3, 3, 4 to win a free bracelet.

And, Customers who spin a 4 win.

Now, We have;

Total outcomes = 8

And, Possible outcomes for who spin a 4 win is,

⇒ 1

Hence, The value of probability that a customer wins a prize is,

⇒ 1 / 8

⇒ 1/8 x 100%

⇒ 12.5%

Thus, The value of probability that a customer wins a prize is,

⇒ 12.5%

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A.Hence proved dy/dx = (2x - 5y)/(5x + 2y). B. The slope of the tangent line at any point on the curve when x=2 is given by (4-5y)/(10+2y). C. The curve has a vertical tangent line at x = 5/√29. D. The x-axis is increasing at a rate of 60 square units per unit time at time t=3. D. The value of da/dt at time t=3 is 60.

A. To show that dy/dx = (2x-5y)/(5x+2y), we differentiate the given equation with respect to x using implicit differentiation:

2x - 2y(dy/dx) - 5y - 5x(dy/dx) = 0. Simplifying and solving for dy/dx, we get:

dy/dx = (2x - 5y)/(5x + 2y)

B. To find the slope of the line tangent to the curve at each point when x=2, we substitute x=2 into the expression we derived in part A:

dy/dx = (2(2) - 5y)/(5(2) + 2y) = (4-5y)/(10+2y)

C. To find the positive value of x at which the curve has a vertical tangent line, we need to find where the slope dy/dx becomes infinite. This occurs when the denominator of dy/dx equals zero, which is when: 5x + 2y = 0

Solving for y in terms of x, we get:

y = (-5/2)x

Substituting this into the equation for the curve, we get:

[tex]x^2 - (-5/2)x^2 - 5x(-5/2)x = 25[/tex]

Simplifying and solving for x, we get:

[tex]x = 5/√29[/tex]

or

[tex]x = -5/√29[/tex]

D. To find the value of da/dt at time t=3, we first use the chain rule to get:

2x(dx/dt) - 2y(dy/dt) - 5y(dx/dt) - 5x(dy/dt) = 0. We are given that x=5, y=0, and dy/dt=-2 when t=3. Substituting these values into the equation above and solving for dx/dt, we get:

dx/dt = (5dy/dt)/(2x-5y) = -10/25 = -2/5 Substituting these values into the expression for da/dt, we get:

[tex]da/dt = 2(5)^2 - 2(0)^2 - 5(0)(-2/5) - 5(5)(-2) = 60[/tex]

So the value of da/dt at time t=3 is 60. This means that the area enclosed by the curve.

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The measure of the missing angle in the right triangle rounded to the nearest 10 or the tens Place is 20°.

The figure in the image is a right triangle.

Measure of missing angle = θ

Opposite to angle θ = 8

Adjacent to angle θ  = 20

To solve for the missing angle, we use the trigonometric ratio.

Note that: tangent = opposite / adjacent

Hence:

tangent θ = opposite / adjacent

tan(θ) = 8/20

tan(θ) = 2/5

Take the tan inverse

θ = tan⁻¹( 2/5 )

θ = 21.8014°

Rounding to the nearest 10 or the tens Place.

θ = 20°

Therefore, the missing angle is 20°.

Option C) 20° is the correct answer.

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The critical value is -1.645 and we reject the null hypothesis at the 0.05 level of significance.

a. To find the critical value(s), we need to use a z-table. Since the alternative hypothesis is one-tailed (p<0.57), we will use the one-tailed z-table. At a significance level of 0.05, the critical value is the z-score that corresponds to an area of 0.05 in the tail of the distribution. From the z-table, we find that the critical value is -1.645.

b. To determine whether to reject or fail to reject the null hypothesis (H), we compare the test statistic (z=-2.12) to the critical value (-1.645).

Since the test statistic is smaller (more negative) than the critical value, we reject the null hypothesis at the 0.05 level of significance. This means that we have sufficient evidence to conclude that the true proportion (p) is less than 0.57.

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The median wait time for the speed slide is 2 times longer than the median wait time for the wave machine.

No, There is a lot of overlap between the two data sets.

We have,

Box plots:

Speed side

Median = 11

First quartile = 6

Third quartile = 12

IQR = 12 - 6 = 6

Wave machine

Median = 9

First quartile = 8

Third quartile = 14

IQR = 14 - 8 = 6

Now,

Difference between the median.

= 11 - 9

= 2

Now,

From the box plots, the wait time for the wave machine is longer than the speed side.

Thus,

The median wait time for the speed slide is 2 times longer than the median wait time for the wave machine.

No, There is a lot of overlap between the two data sets.

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To use the unit circle to find the exact value of sin(330°), we can follow these steps:

        330° - 300° = 30°

Since the reference angle is 30°, the corresponding point is located on the terminal side of the angle formed by rotating 30° counterclockwise from the positive x-axis. This point has coordinates of (cos(30°), sin(30°)), which are (√3/2, 1/2).

Use the sign of the trig function in the appropriate quadrant to determine the final value of sin(330°). Since 330° is in the fourth quadrant and the sine function is negative in the fourth quadrant, sin(330°) = -sin(30°) = -1/2.

Therefore, the exact value of sin(330°) is -1/2.

The sum of money that will grow to $2,324.61 in two years at a 4% interest rate compounded quarterly is $2,145.00.

Sum of money that will grow to $2,324.61 in two years at a 4% interest rate compounded quarterly, we will use the formula for compound interest:

Future Value = Principal * (1 + (Interest Rate / Number of Compounds))^ (Number of Compounds * Time)

Here, we need to find the Principal amount. The given values are:
- Future Value = $2,324.61
- Interest Rate = 4% = 0.04
- Number of Compounds per year = 4 (quarterly)
- Time = 2 years

Rearranging the formula to find the Principal:

Principal = Future Value / (1 + (Interest Rate / Number of Compounds))^ (Number of Compounds * Time)

Substitute the values into the formula:

Principal = 2324.61 / (1 + (0.04 / 4))^(4 * 2)

Principal = 2324.61 / (1 + 0.01)^8
Principal = 2324.61 / (1.01)^8
Principal = 2324.61 / 1.082857169

Principal = $2,145.00 (rounded to the nearest cent)

The sum of money that will grow to $2,324.61 in two years at a 4% interest rate compounded quarterly is $2,145.00.

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The projection of vector u in the direction of vector v is equal to P = - 6.354 i - 5.718 j.

In this problem we need to determine the expression of the projection of vector u in the direction of vector v, whose formula is now introduced:

P = [(u • v) / ||v||²] · v

Where:

If we know that u = - 7 i - 5 j and v = - 10 i - 9 j, then the projection of the vector is:

u • v = 70 + 45

u • v = 115

||v||² = 100 + 81

||v||² = 181

P = (115 / 181) · (- 10 i - 9 j)

P = - (1150 / 181) i - (1035 / 181) j

P = - 6.354 i - 5.718 j

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The number of MAD's that represents the difference of the means of each data-set is given as follows:

C. 0.2

The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.

Hence, for the first period, the mean is obtained as follows:

Mean = (3 x 0 + 4 x 1 + 5 x 2 + 2 x 3 + 1 x 4)/(3 + 4 + 5 + 2 + 1)

Mean = 1.6.

For the second period, the mean is obtained as follows:

Mean = (4 x 0 + 5 x 1 + 4 x 2 + 0 x 3 + 2 x 4)/(4 + 5 + 4 + 0 + 2)

Mean = 1.4.

The difference is then given as follows:

1.6 - 1.4 = 0.2 -> which is 0.2 MADs, as MAD = 1.

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The matrix 2x2 of A is:

[20  30]
[ 0 -15]

What is the eigenvector?

The eigenvector is a vector that is associated with a set of linear equations. The eigenvector of a matrix is also known as a latent vector, proper vector, or characteristic vector. These are defined in the reference of a square matrix.

To solve this problem, we need to use the fact that a matrix can be diagonalized only if it has a set of linearly independent eigenvectors.

First, we need to find the eigenvectors and eigenvalues of the matrix. Let A be the matrix we want to find.

We know that [2 3] is an eigenvector of A with eigenvalue 10, so we have:

A[2 3] = 10[2 3]

Multiplying out the matrices, we get:

[2 3] [a b] = [20 30]

where a and b are the unknown entries of A. Solving this system of equations, we get a = 5 and b = 10. Therefore, the matrix A is:

[5 10]
[0 3]

Now, we need to check if [0 3] is also an eigenvector of A with eigenvalue -5:

A[0 3] = -5[0 3]

[5 10] [0 3] = [0 -15]

Multiplying out the matrices, we get:

[0 30] = [0 -15]

This is a contradiction since the two matrices are not equal. Therefore, [0 3] is not an eigenvector of A with eigenvalue -5.

In summary, the matrix A that satisfies the given conditions is:

[5 10]
[0 3]

with eigenvectors [2 3] and [0 1] and eigenvalues 10 and 3, respectively.
To find a 2x2 matrix with eigenvectors [2, 3] and [0, 3] and eigenvalues 10 and -5, respectively, follow these steps:

Step 1: Associate the eigenvectors with their respective eigenvalues:
- Eigenvector [2, 3] has eigenvalue 10.
- Eigenvector [0, 3] has eigenvalue -5.

Step 2: Write the eigenvalue-eigenvector equations:
- 10 * [2, 3] = A * [2, 3]
- (-5) * [0, 3] = A * [0, 3]

Step 3: Expand the equations:
- 10 * [2, 3] = [20, 30]
- (-5) * [0, 3] = [0, -15]

Step 4: Create the matrix A using the expanded equations:
A = [20, 30; 0, -15]

So, the 2x2 matrix A is:
[20  30]
[ 0 -15]

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Yes, I do agree that Kelly can't put a right triangle in either of the groups because it does not have two pairs of parallel sides.

In Mathematics and Geometry, a right angle can be defined as a type of angle that is formed in a triangle by the intersection of two (2) straight lines at 90 degrees. This ultimately implies that, a right angled triangle has a measure of 90 degrees.

Based on the Venn diagram shown in the image attached below, we can reasonably infer and logically deduce that Kelly was correct by saying can't put a right triangle or right angled triangle in either of the groups because it does not have two pairs of parallel sides.

However, Kelly can put a square or rectangle in either of the groups because they have two pairs of parallel sides.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

  4 ft³

Step-by-step explanation:

Given similar pyramids with heights 12 ft and 4 ft, you want the volume of the smaller when the volume of the larger is 256 ft³.

The scale factor for volume is the cube of the scale factor for linear dimensions. The height of the smaller pyramid is 1/4 the height of the larger, so its volume will be (1/4)³ = 1/64 times that of the larger.

The volume of Pyramid B is (1/64)(256 ft³) = 4 ft³.

Answer: 39

Step-by-step explanation:

    First, we will list some multiples of 3.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, [tex]\boxed{39}[/tex], 42, 45, 48, 51, 54, etc.

    Next, we will list some multiples of 13.

13, 26, [tex]\boxed{39}[/tex], 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, etc.

    We see that the least common multiple of 3 and 13 is 39. This is the smallest value that shows up in both lists.

    What is a multiple?

A multiple is a number that can be divided with that number without a reminder.

[tex](Ax^2 + 4Ax + 2B)e^x + (2A + B + Ce^x) - 2((Ax^2 + 2Ax + B)e^x + (B + Ce^x)) - 3(Ax^2e^x + Bxe^x + Ce^[/tex]

We can use the method of undetermined coefficients to find particular solutions to these

differential equations.

For y" - [tex]3y' + 2y = (e^3x (1 + x)[/tex], we assume a particular solution of the form y_p = Ae^3x(1 + x) + Bx^2 + Cx + D. Then, [tex]y_p' = 3Ae^3x(1 + x) + 2Bx + C[/tex]and y_p" [tex]= 9Ae^3x + 2B[/tex]. Substituting these into the differential equation, we get:

[tex]9Ae^3x + 2B - 9Ae^3x - 6Ae^3x - 3Ae^3x + 3Ae^3x(1 + x) + 2Bx + Cx + D = e^3x(1 + x)[/tex]

Simplifying and collecting like terms, we get:

[tex](3A + 2B)x + Cx + D = e^3x(1 + x)[/tex]

Matching coefficients, we have:

3A + 2B = 0

C = 1

D = 0

Solving for A and B, we get:

A = -2/9

B = 3/4

Therefore, a particular solution is [tex]y_p = (-2/9)e^3x(1 + x) + (3/4)x^2 + x[/tex].

For [tex]y" - 6y' + 5y = e^-3[/tex]x([tex]35 - 8x[/tex]), we assume a particular solution of the form [tex]y_p = Ae^-3x + Bx + C[/tex]. Then, [tex]y_p' = -3Ae^-3x + B[/tex] and [tex]y_p" = 9Ae^-3x[/tex]. Substituting these into the differential equation, we get:

[tex](9A + B)x + (-6A - 6B + C) = e^-3x(35 - 8x[/tex]

Simplifying and collecting like terms, we get:

[tex](9A + B)x + (-6A - 6B + C) = e^-3x(35 - 8x[/tex])

Matching coefficients, we have:

9A + B = 0

-6A - 6B + C = 35

Solving for A, B, and C, we get:

A = -5/27

B = 15/27 = 5/9

C = 290/27

Therefore, a particular solution is y_p [tex]= (-5/27)e^-3x + (5/9)x + 290/27.For y" - 2y' - 3y = e^x[/tex] [tex](-8 + 3x)[/tex], we assume a particular solution of the form [tex]y_p = Ax^2e^x + Bxe^x + Ce^x. Then, y_p' = (Ax^2 + 2Ax + B)e^x + (B + Ce^x) and y_p" = (Ax^2 + 4Ax + 2B)e^x + (2A + B + Ce^x)[/tex]. Substituting these into the differential equation, we get:

[tex](Ax^2 + 4Ax + 2B)e^x + (2A + B + Ce^x) - 2((Ax^2 + 2Ax + B)e^x + (B + Ce^x)) - 3(Ax^2e^x + Bxe^x + Ce^[/tex]

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