I need help with C and D can someone help me please

a. The probability that X is greater than 500 is approximately 0.368.

b. The conditional probability that X is greater than 1000 is approximately 0.368.

a. To compute Pr[X > 500), we use the cumulative distribution function (CDF) of the exponential distribution, which is:

F(x) = 1 - e^(-x/B)

Plugging in B = 500 and x = 500, we get:

Pr[X > 500) = 1 - F(500) = 1 - (1 - e^(-500/500)) = e^(-1) ≈ 0.368

Therefore, the probability that X is greater than 500 is approximately 0.368.

b. To compute Pr[X > 1000 | X > 500), we use the definition of conditional probability:

Pr[X > 1000 | X > 500) = Pr[(X > 1000) ∩ (X > 500)] / Pr[X > 500)

Since X is a continuous random variable, we can rewrite the probability of the intersection using the minimum of X:

Pr[(X > 1000) ∩ (X > 500)] = Pr[X > max(1000, 500)] = Pr[X > 1000]

Plugging in B = 500 into the CDF, we have:

Pr[X > 1000] = 1 - F(1000) = 1 - (1 - e^(-1000/500)) = e^(-2) ≈ 0.135

We already know from part a that Pr[X > 500) = e^(-1) ≈ 0.368.

Putting it all together, we have:

Pr[X > 1000 | X > 500) = Pr[X > 1000] / Pr[X > 500) = (e^(-2)) / (e^(-1)) = e^(-1) ≈ 0.368

This result shows that the conditional probability that X is greater than 1000, given that it is already greater than 500, is the same as the probability that X is greater than 500 on its own. In other words, knowledge of the fact that X is already greater than 500 does not change our prediction about whether it will be greater than 1000 or not.

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Answer:

125 questions

Answer:

125 questions

Step-by-step explanation:

The number of one-to-one functions from a set with five elements to sets with one, two, three, four, and five elements are 1, 20, 60, 120, and 1, respectively.To answer this question, we need to use the concept of one-to-one functions.

A one-to-one function is a function where each element in the domain corresponds to a unique element in the range. In other words, no two elements in the domain can have the same image in the range.

Let's consider each case separately.

1. Set with one element: In this case, there is only one possible function since there is only one element in the range that needs to be mapped to.

2. Set with two elements: There are a total of 20 possible one-to-one functions from a set with five elements to a set with two elements. To see why, we can think of it as choosing two distinct elements from the domain to map to the two elements in the range. There are 5 choices for the first element, and 4 choices for the second element (since we can't choose the same element twice). So the total number of possible functions is 5 x 4 = 20.

3. Set with three elements: There are a total of 60 possible one-to-one functions from a set with five elements to a set with three elements. To see why, we can think of it as choosing three distinct elements from the domain to map to the three elements in the range. There are 5 choices for the first element, 4 choices for the second element, and 3 choices for the third element. So the total number of possible functions is 5 x 4 x 3 = 60.

4. Set with four elements: There are a total of 120 possible one-to-one functions from a set with five elements to a set with four elements. To see why, we can think of it as choosing four distinct elements from the domain to map to the four elements in the range. There are 5 choices for the first element, 4 choices for the second element, 3 choices for the third element, and 2 choices for the fourth element. So the total number of possible functions is 5 x 4 x 3 x 2 = 120.

5. Set with five elements: In this case, there is only one possible function since there are five elements in both the domain and the range, and every element in the domain must be mapped to a unique element in the range.

In summary, the number of one-to-one functions from a set with five elements to sets with one, two, three, four, and five elements are 1, 20, 60, 120, and 1, respectively.

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The sum of the series [tex]\sum_{k=0}^\infty \frac{3}{7^{k} }[/tex]  is 7/2. Therefore, the correct answer is option C. The sum of a geometric series can be found only if the ratio is between -1 and 1.

To find the sum of the series [tex]\sum_{k=0}^\infty \frac{3}{7^{k} }[/tex], we can use the formula for the sum of an infinite geometric series, which is [tex]\frac{a}{1-r}[/tex], where a is the first term and r is the common ratio.

In this case, the first term is [tex]\frac{3}{7^0}=3[/tex] and the common ratio is [tex]\frac{1}{7}[/tex]. Substituting these values into the formula, we get:

[tex]\frac{3}{1-\frac{1}{7}}=\frac{3}{\frac{6}{7}}=\frac{7}{2}[/tex]

Therefore, the sum of the series is c. 7/2. Alternatively, we can also find the sum of the series by adding up the terms:

[tex]\frac{3}{1}+\frac{3}{7}+\frac{3}{49}+\frac{3}{343}+...\approx 4.5[/tex]

This method involves adding up an infinite number of terms, so it may not always be practical or accurate. Using the formula for the sum of an infinite geometric series is a more efficient and reliable method. Therefore, the correct answer is option C.

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Complete Question:

Find the sum of the series:

[tex]\sum_{k=0}^\infty \frac{3}{7^{k} }[/tex]

a. 7/3

b. 21/2

c. 7/2

d. 21/4

e. 7

Based on the given percentiles, we can determine that the value of X represents the resting heart rate at the 25th percentile. To estimate the value of X, we can use the z-score formula for a normal distribution. The z-score corresponding to the 25th percentile is approximately -0.674.

The formula for the z-score is: (X - mean) / standard deviation. We have the mean (70 bpm) and the z-score (-0.674), but we need to estimate the standard deviation. To do this, we can use the 50th and 75th percentiles. The z-score for the 50th percentile is 0, so the mean equals the 50th percentile value (70 bpm). The z-score for the 75th percentile is 0.674, so we can set up the equation:

(80 - 70) / standard deviation = 0.674

10 / standard deviation = 0.674

Standard deviation ≈ 10 / 0.674 ≈ 14.8 bpm

Now we can solve for X:

-0.674 = (X - 70) / 14.8

-0.674 * 14.8 ≈ X - 70

-9.9 ≈ X - 70

X ≈ 60.1

Rounding to the nearest whole number, the value of X is approximately 60 bpm.

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The white cylindrical silo has a diameter of 30 feet, which means the radius is 15 feet. The height of the silo is 80 feet. The area of the red stripe on the cylindrical silo is approximately 565.5 square feet.

A red stripe with a horizontal width of 3 feet is painted on the silo, making two complete revolutions around it. This means the total length of the red stripe is 2 times the circumference of the base of the silo plus 2 times the circumference of the top of the silo. The circumference of the base of the silo is 2 times pi times the radius, which is 2 x 3.14 x 15 = 94.2 feet.
The circumference of the top of the silo is also 94.2 feet
So the total length of the red stripe is 2 x 94.2 + 2 x 94.2 = 376.8 feet.  The horizontal width of the red stripe is 3 feet, so the area of the red stripe is 376.8 x 3 = 1130.4 square feet.
To find the total surface area of the silo, we need to find the area of the two circular ends and the area of the curved surface. The area of each circular end is pi times the radius squared, which is 3.14 x 15 x 15 = 706.5 square feet.
The area of the curved surface is the product of the height, the circumference, and 2 (since there are two sides), which is 80 x 94.2 x 2 = 15,088 square feet.
So the total surface area of the silo is 2 x 706.5 + 15,088 = 15,501 square feet.
Therefore, the area of the red stripe as a percentage of the total surface area of the silo is (1130.4/15,501) x 100% = 7.29%.

A white cylindrical silo has a diameter of 30

feet and a height of 80

feet. A red stripe with a horizontal width of 3

feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

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Answer:24

Step-by-step explanation:

If you do 72 divided by 3 you get 24

It will be impossible to know if the photographer has enough space left on her memory card without knowing the capacity of the card and the size of the planned photos for session D.

Main answer:

It is impossible to determine if the photographer has enough space left on her memory card without knowing the capacity of the card and the size of the planned photos for session D.

We must know capacity of the card and the size of the planned photos for session D. If combined size of the planned photos for session B and C is less than remaining space on the card after accounting for the 3.4 megabytes needed for session D, then, the photographer would have enough space.

But if combined size of the planned photos for session B and C is greater than the remaining space on the card after accounting for the 3.4 megabytes needed for session D, then, the photographer would not have enough space.

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Answer: Dome Fuji, in Antarctica, is the coldest place in the world with a record low of –92 degrees.

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

a/2 = 36/18

Reduce the right fraction.

a/2 = 2/1

Multiply both sides by 2.

a = 4

Answer: 4

An expression that represent the inverse of the matrix below include the following: D. [tex]\frac{1}{5} \left[\begin{array}{ccc}2&-3\\1&1\end{array}\right][/tex]

In Mathematics, an inverse function simply refers to a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).

In this exercise, you are required to determine the inverse of the matrix below. This ultimately implies that, we would determine the determinant of the matrix as follows;

Determinant of A = detA = (1 × 2) - (-1)(3)

Determinant of A = detA = 2 - (-3)

Determinant of A = detA = 2 + 3

Determinant of A = detA = 5 ≠ 0

Since the determinant of A is not equal to zero (detA ≠ 0), we can logically deduce that, the inverse of A (A⁻¹) exist;

Adj(A) = [tex]\left[\begin{array}{ccc}2&-3\\1&1\end{array}\right][/tex]

A⁻¹ = [tex]\frac{1}{detA}[/tex][Adj(A)]

A⁻¹ =     [tex]\frac{1}{5} \left[\begin{array}{ccc}2&-3\\1&1\end{array}\right][/tex]

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Each irrational number with the number line:

a) First number line = 2.1829391321..

b) Second number line = 2.364125..

c) Third number line = 2.18112..

d) Fourth number line = 2.1823912..

a) Here, we can obseve that between 2.182 and 2.183 a number line is divided into 10 equal segments.

So, each unit length represents (2.183 - 2.182) / 10 = 0.0001 unit

The required irrational number (represented by red dot) lie between 2.1829 and 2.183.

So, the  irrational number would be 2.1829391321..

b) Here, we can obseve that between 2 and 3 a number line is divided into 10 equal segments.

So, each unit length represents (3 - 2) / 10 = 0.1 unit

The required irrational number (represented by red dot) lie between 2.3 and 2.4

So, the irrational number would be 2.364125..

c) Here, we can obseve that between 2.11 and 2.21 a number line is divided into 10 equal segments.

So, each unit length represents (2.21 - 2.11) / 10 = 0.01 unit

The required irrational number (represented by red dot) lie between 2.18 and 2.19

So, the irrational number would be 2.18112..

d) Here, we can obseve that between 2.18 and 2.19 a number line is divided into 10 equal segments.

So, each unit length represents (2.19 - 2.18) / 10 = 0.001 unit

The required irrational number (represented by red dot) lie between 2.182 and 2.183

So, the irrational number would be 2.1823912..

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The x-coordinate where the tree will be planted in a circular sidewalk constructed around a park, with a brick pathway through its diameter from (-1,4) to (9,10), is 4, which is the x-coordinate of the center of the circle.

To find the center of the circle, we need to find the midpoint of the diameter. We can use the midpoint formula

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

where (x₁, y₁) and (x₂, y₂) are the endpoints of the diameter.

Using the coordinates (-1, 4) and (9, 10), we get

Midpoint = ((-1 + 9)/2, (4 + 10)/2) = (4, 7)

So the center of the circle is at the point (4, 7).

Since the tree will be planted at the center of the circle, its x-coordinate will be the same as the x-coordinate of the center of the circle, which is 4.

Therefore, the x-coordinate where the tree will be planted is 4.

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--The given question is incomplete, the complete question is given

" a circular sidewalk is being constructed around the perimeter of a local park a brick pathway will be added through the diameter of the circle is from (-1,4) and (9,10) coordinate plane and a tree will be planted in the sidewalk at the center of the circle what is the x coordinate where the tree will be planted "--

The largest amount of money she could have is: (3 x 25 cents) + (3 x 10 cents) + (2 x 5 cents) = 75 cents + 30 cents + 10 cents = 115 cents or $1.15.

To find the largest amount of money Tasha could have with 8 coins in her pocket, we need to consider the different combinations of coins she could have. Since she has no more than 3 of any coin, the possibilities are:

- 3 quarters, 2 dimes, 1 nickel, 2 pennies = $0.81
- 3 quarters, 2 dimes, 2 nickels, 1 penny = $0.80
- 3 quarters, 2 nickels, 3 pennies = $0.78
- 3 quarters, 1 dime, 3 nickels, 1 penny = $0.76
- 3 quarters, 1 dime, 2 nickels, 3 pennies = $0.74
- 3 quarters, 1 dime, 1 nickel, 4 pennies = $0.73
- 2 quarters, 3 dimes, 1 nickel, 2 pennies = $0.70
- 2 quarters, 3 dimes, 2 nickels, 1 penny = $0.69
- 2 quarters, 2 dimes, 3 nickels, 1 penny = $0.68
- 2 quarters, 2 dimes, 2 nickels, 2 pennies = $0.67

Therefore, the largest amount of money Tasha could have is $0.81 with 3 quarters, 2 dimes, 1 nickel, and 2 pennies.

To maximize the amount of money Tosha could have with 8 coins and no more than 3 of any coin, she should carry the coins with the highest denominations. In this case, she can have 3 quarters (25 cents each), 3 dimes (10 cents each), and 2 nickels (5 cents each). The largest amount of money she could have is:

(3 x 25 cents) + (3 x 10 cents) + (2 x 5 cents) = 75 cents + 30 cents + 10 cents = 115 cents or $1.15.

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A differential equation is an equation that relates an unknown function to its derivatives, or differentials, with respect to one or more independent variables. Therefore, N(t) = 666.67 - 566.67 * exp(-0.15*t + 0.7102)

The differential equation model for the population size N at time t is:

dN/dt = (birth rate - death rate + immigration rate) * N

where birth rate = per capita birth rate * N, death rate = per capita death rate * N, and immigration rate = constant immigration rate = 100.

Substituting the given values, we get:

dN/dt = (0.15N - 0.3N + 100) * N

dN/dt = (-0.15*N + 100) * N

dN / (-0.15*N + 100) = dt

-6.6667 ln(-0.15*N + 100) = t + C

where C is the constant of integration.

N(t) = (100/0.15) - (100/0.15) * exp(-0.15t - 6.6667C)

where (100/0.15) = 666.67.

To determine the value of the constant C, we need an initial condition. Let's assume that the initial population size N(0) = 1000. Substituting this in the above equation, we get:

1000 = 666.67 - (666.67 * exp(-6.6667*C))

Solving for C, we get: C = -0.1057

Substituting this value of C in the equation for N(t), we get:

N(t) = 666.67 - 566.67 * exp(-0.15*t + 0.7102)

Therefore, the model for the population size N at time t, reflecting the given assumptions, is:

N(t) = 666.67 - 566.67 * exp(-0.15*t + 0.7102)

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The ratio that is equivalent to 4:6 is 40:60

A ratio is a mathematical expression of comparing two similar or different quantities by division.

For examples if the ratio of cow to sheep In a farm is 3: 4, this means that for 3 cows in the farm there will be 4 sheeps

Equivalent ratios are the ratios that are the same when we compare them. Examples of equivalent ratios are 4:5 and 8 :10.

The equivalent of 4:6 can also be 2:3 but in the options we don't have that, another equivalent can be obtained by multiplying 10 to both sides

=4×10: 6× 10

= 40:60

therefore the equivalent of the ratio 4:6 is 40:60

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If the mean cost of new books is the same as the median cost of new books, then the distribution of book costs is symmetrically distributed.

This means that the data is evenly distributed around the mean and median, and there are an equal number of values on both sides of the central point. In a symmetric distribution, the mean and median are the same, and the data is evenly spread out around them.

A negatively skewed distribution would have a longer tail on the left side, indicating that the majority of the values are higher. A positively skewed distribution would have a longer tail on the right side, indicating that the majority of the values are lower.

A symmetric distribution, on the other hand, has no long tail on either side, and the majority of the values cluster around the central point. Therefore, the distribution of book costs is symmetrically distributed.

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The number of miles that Freya traveled in total is 320 miles.

Given that:

Bournemouth to Gloucester: v = 50 mph and t = 2.5 h

Gloucester to Anglesey: v = 65 mph and t = 3 h

We know that the speed formula

Speed = Distance/Time

The number of miles that Freya traveled in total is calculated as,

Distance = 50 x 2.5 + 65 x 3

Distance = 125 + 195

Distance = 320 miles

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Cross multiplying

20 × 5 = t × t

100 = 2t

Dividing 2 on the opposite side

100/2 = t

t = 50

The total number of milliliters of perfume is 1,600 mL. Then the correct option is D.

Two cases of perfume are delivered to a retailer. There are ten bottles in each case. There are 80 millimeters of perfume in each bottle.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The total number of milliliters of perfume is calculated as,

⇒ 2 x 10 x 80

⇒ 1,600 mL

Thus, the correct option is D.

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To find the outward flux, we can use the divergence theorem, which states that the flux of a vector field through a closed surface S is equal to the volume integral of the divergence of the vector field over the region enclosed by S.

In this problem, the vector field is F = (tan y + (8y + 3z)i + 2z + 8 cos x) + (Vx^2 + y^2 + 22k). The surface S is the region bounded by the graphs of z = Vx^2 + y^2 and x^2 + y^2 +22 = 9.

To apply the divergence theorem, we first need to find the divergence of the vector field. Using the product and chain rules, we have:

div F = (∂/∂x)(tan y + (8y + 3z)) + (∂/∂y)(2z + Vx^2 + y^2 + 22) + (∂/∂z)(Vx^2 + y^2 + 22)

Simplifying each term, we get:

div F = 8 + 2Vx + 2y

Next, we need to find the volume enclosed by S. This can be done by integrating the equation of the sphere and the equation of the cylinder over their respective domains:

V = ∫∫∫ dV = ∫∫ dz dA = ∫∫ (9 - x^2 - y^2)^(1/2) dA

where the limits of integration are:

-3 ≤ x ≤ 3
-(9-x^2)^(1/2) ≤ y ≤ (9-x^2)^(1/2)

We can now apply the divergence theorem:

flux = ∫∫ F · dS = ∫∫∫ div F dV = ∫∫ dz dA div F

Using the limits of integration for V and A, we get:

flux = ∫∫ (9 - x^2 - y^2)^(1/2) dA (8 + 2Vx + 2y)

Using polar coordinates for A, we have:

flux = ∫0^2π ∫0^3 (9 - r^2)^(1/2) r dr dθ (8 + 2r cos θ + 2r sin θ)

Simplifying and evaluating the integral, we get:

flux = 216π

Therefore, the outward flux of the vector field F through the surface S is 216π.

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we can expect to receive approximately 3 phone calls during the 1.5-hour movie, on average.

The probability of your phone ringing during a 1.5-hour movie can be calculated using the Poisson distribution formula:

P(X = k) = (e^-λ * λ^k) / k!

Where X is the number of phone calls, λ is the average rate of calls per unit time (in this case, 2 calls per hour), and k is the number of calls during the 1.5-hour period.

So, for k = 0 (no calls), the probability is: P(X = 0) = (e^-2 * 2^0) / 0! = e^-2 ≈ 0.1353

Therefore, the probability that your phone rings at least once during the movie is: P(X ≥ 1) = 1 - P(X = 0) = 1 - e^-2 ≈ 0.8647

To calculate the expected number of calls during the movie, we use the formula: E(X) = λ * t

Where t is the duration of the period (1.5 hours in this case). So, the expected number of calls during the movie is: E(X) = 2 * 1.5 = 3

Therefore, we can expect to receive approximately 3 phone calls during the 1.5-hour movie, on average.

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Answer:

31.7925 [tex]in^{3}[/tex]

Step-by-step explanation:

V = 1/3[tex]\pi r^{2}[/tex]h

v = 1/3 (3.14)([tex]2.25^{2}[/tex])(6)    The radius is 1/2 of the diameter

v = 1/3 (3.14)(5.0625)(6)

v = [tex]\frac{95.3775}{3}[/tex]

v = 31.7925

Helping in the name of Jesus.

The solution to the initial value problem is:

u(x,t) = 6 sin(9πx/L) exp(-81kπ^2 t/L^2) + 3/2 sin(πx/L) exp(-kπ^2 t/L^2) - 1/2 sin(3πx/L) exp(-9kπ^2 t/L^2) + 2 cos(3πx/L) exp(-9kπ^2 t/L^2)

To solve the heat equation, we can use separation of variables method assuming that the solution can be written as a product of functions of x and t, i.e.,

u(x,t) = X(x)T(t)

Then, the heat equation becomes:

X(x)T'(t) = kX''(x)T(t)

Dividing both sides by kX(x)T(t) and rearranging, we get:

1/k * T'(t)/T(t) = X''(x)/X(x) = -λ

where λ is a constant.

We can then solve for X(x) and T(t) separately:

X''(x) + λX(x) = 0

The boundary conditions u(0,t) = u(L,t) = 0 give X(0) = X(L) = 0, which leads to the solution:

X(x) = B sin(nπx/L)

where n = 1,2,3,... and B is a constant.

Using the initial conditions, we can determine the coefficients B_n for each value of n:

u(x,0) = 6 sin(9πx/L) = B_9 sin(9πx/L)

So, B_9 = 6.

u(x,0) = 3 sin(πx/L) - sin(3πx/L) = B_1 sin(πx/L) - B_3 sin(3πx/L)

Solving for B_1 and B_3, we get:

B_1 = 3/2, B_3 = -1/2

u(x,0) = 2 cos(3πx/L) = B_3 cos(3πx/L)

So, B_3 = 2.

Now, we can solve for T(t) using T'(t)/T(t) = -kλ. This leads to the solution:

T(t) = C exp(-kλt)

where C is a constant.

Finally, we can write the solution to the heat equation as:

u(x,t) = ∑ B_n sin(nπx/L) exp(-k(nπ/L)^2 t)

Substituting the values of B_n for each initial condition, we get:

u(x,t) = 6 sin(9πx/L) exp(-81kπ^2 t/L^2) + 3/2 sin(πx/L) exp(-kπ^2 t/L^2) - 1/2 sin(3πx/L) exp(-9kπ^2 t/L^2) + 2 cos(3πx/L) exp(-9kπ^2 t/L^2)

Therefore, the solution to the initial value problem is:

u(x,t) = 6 sin(9πx/L) [tex]e^{-81kπ^2 t/L^2}[/tex] + 3/2 sin(πx/L)[tex]e^{-kπ^2 t/L^2}[/tex] - 1/2 sin(3πx/L) [tex]e^{-9kπ^2 t/L^2}[/tex] + 2 cos(3πx/L) [tex]e^{-9kπ^2 t/L^2}[/tex]

where k is the thermal diffusivity of the material.

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The variance in the dependent variable can be explained by the variance in the independent variable 66.67%.

To find the variance in the dependent variable that can be explained by the variance in the independent variable, we need to first calculate the coefficient of determination (R-squared).

The R-squared value is a statistical measure that determines the proportion of the variation in the dependent variable that can be explained by the independent variable.
R-squared is calculated as the ratio of the explained variation (SSR) to the total variation (SST).

Therefore, we can calculate the R-squared as follows:
R-squared = SSR/SST = 24/36 = 0.67
This means that 67% of the variation in the dependent variable can be explained by the variation in the independent variable.
To find the variance in the dependent variable that can be explained by the variance in the independent variable, we need to multiply the R-squared value by the total variance in the dependent variable (SST).

Therefore, we can calculate the variance explained by the independent variable as follows:
Variance explained = R-squared * SST = 0.67 * 36 = 24.12
Therefore,

The variance in the dependent variable that can be explained by the variance in the independent variable is 24.12.

This means that the independent variable can explain 24.12 units of variation in the dependent variable, while the remaining 11.88 units of variation are due to other factors not accounted for by the independent variable.

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You would be paying $3000 per year for the health insurance plan.

If the premium for the health insurance plan is $125 per paycheck, and you get paid every two weeks, then the cost of the insurance per month is:

2 paychecks x $125 per paycheck = $250 per month

To find the cost per year, we need to multiply the monthly cost by 12:

$250 per month x 12 months = $3,000 per year

hence, you would be paying $3000 per year for the health insurance plan.

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The equation of the linear relationship in slope-intercept form is expressed as: y = 3x - 4.

To find the equation, pick two points on the line and find the slope (m), then find the y-intercept (b) of the line.

Using, (0, -4) and (1, -1):

Slope (m) = (-1 -(-4)) / (1 - 0)

m = 3/1

m = 3

The y-intercept (b) is -4.

Substitute m = 3 and b = -4 into y = mx + b:]

y = 3x - 4

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The solution to the boundary value problem is

y(x) = -(2 - 2e^3)e^3/(1 - e^3) + (2 - 2e^3)/(1 - e^3) * e^(3x).

The characteristic equation is r^2 - 3r = 0, which has roots r = 0 and r = 3. Thus, the general solution to the differential equation is y(x) = c1 + c2e^(3x).

Using the initial condition y(0) = 2 - 2e^3, we have y(0) = c1 + c2 = 2 - 2e^3.

Using the boundary condition y(1) = 0, we have y(1) = c1 + c2e^3 = 0.

We can solve for c1 and c2 by solving the system of equations:

c1 + c2 = 2 - 2e^3

c1 + c2e^3 = 0

Subtracting the second equation from the first, we get c2(1 - e^3) = 2 - 2e^3, which gives us c2 = (2 - 2e^3)/(1 - e^3).

Substituting c2 into the second equation, we get c1 = -c2e^3 = -(2 - 2e^3)e^3/(1 - e^3).

Thus, the solution to the boundary value problem is y(x) = -(2 - 2e^3)e^3/(1 - e^3) + (2 - 2e^3)/(1 - e^3) * e^(3x).

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