6. Katya and Alyse are sewing a border with
yarn on a rectangular blanket. The length of the
long side is 6 feet and the short side is 4 feet.
How many feet of yarn will they need?

Answer: The formula for the surface area of a sphere is given by 4πr^2, and since we have half of a sphere, the surface area of a dome (a half sphere) is 2πr^2.

Plugging in the radius r = 12 meters, we get:

Surface area = 2π(12)^2

Surface area = 2π(144)

Surface area ≈ 904.78

Rounding to the nearest whole number, we get the surface area of the dome as 905 meters squared. Therefore, the closest answer choice is 967 meters squared.

So the answer is: 967 meters squared.

Step-by-step explanation:

The x^6y^7 term in the expansion will be 36x^2y^7.

The x^7y^2 term in the expansion will be 36x^7y^2.

In the given problem, we have the binomial expansion of (x - y)^n, which includes the term 126x^5y^4.

We will use the binomial coefficient formula to find the coefficients of the desired terms.

The general term of the binomial expansion is given by:

T(k) = C(n, k) * x^(n-k) * y^k

where C(n, k) is the binomial coefficient and can be calculated as:

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

From the given term, 126x^5y^4, we have:

126 = C(n, 4)

x^5 = x^(n-4)

y^4 = y^4

Now we can find the value of n:

126 = n! / (4!(n-4)!)

Let's solve for n:

126 * 4! = n! / (n-4)!

504 = n! / (n-4)!

Now, we will find the coefficients of the terms x^6y^7 and x^7y^2.

The x^6y^7 term in the expansion will be:

T(k) = C(n, 7) * x^(n-7) * y^7

Since x^5 = x^(n-4), we have n - 4 = 5, so n = 9.

Substituting the value of n:

T(k) = C(9, 7) * x^2 * y^7

Using the binomial coefficient formula:

C(9, 7) = 9! / (7!2!) = 36

So, the x^6y^7 term in the expansion will be 36x^2y^7.

The x^7y^2 term in the expansion will be:

T(k) = C(n, 2) * x^(n-2) * y^2

We already found that n = 9, so substituting the value of n:

T(k) = C(9, 2) * x^7 * y^2

Using the binomial coefficient formula:

C(9, 2) = 9! / (2!7!) = 36

So, the x^7y^2 term in the expansion will be 36x^7y^2.

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The truth table for (A ⋁ B) ⋀ ~(A ⋀ B) is:

A   B   (A ⋁ B) ⋀ ~(A ⋀ B)

0   0                0

0    1                0

1     0               0

1     1                0

A truth table is a table that displays all possible combinations of truth values (true or false) for one or more propositions or logical expressions, as well as the truth value of the resulting compound proposition or expression that is created by combining them using logical operators like AND, OR, NOT, IMPLIES, etc.

The columns of a truth table reflect the propositions or expressions themselves as well as the compound expressions created by applying logical operators to them. The rows of a truth table correspond to the various possible combinations of truth values for the propositions or expressions.

To complete the truth table for (A ⋁ B) ⋀ ~(A ⋀ B), we need to consider all possible combinations of truth values for A and B.

A   B   A ⋁ B   A ⋀ B   ~(A ⋀ B)   (A ⋁ B) ⋀ ~(A ⋀ B)

0   0      0            0             1                      0

0    1       1            0             1                      0

1    0       1            0             1                      0

1    1        1             1             0                     0

So, the only case where the expression is true is when both A and B are true, and for all other cases it is false.

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Step-by-step explanation:

From a z-score table

   z-score = 1 corresponds to  .8413   or  84.13 percentile

     meaning 84 .13 % have a lesser score than you

Answer: (a) If you randomly selected one bottle instead of 16, it would be LESS likely to contain a volume of cola within ±1 milliliter of μ.

The reason is that as the sample size increases, the sample mean (x-bar) tends to be a more accurate estimator of the population mean (μ) due to the central limit theorem. With a larger sample size, the sample mean is less likely to deviate significantly from the population mean. In this case, with a sample size of 16 bottles, the probability that x-bar estimates μ to within ±1 milliliter is 0.8176, which means that the sample mean is likely to be within ±1 milliliter of the population mean in about 81.76% of the cases.

On the other hand, if you randomly selected just one bottle instead of 16, the variability of the individual bottle's volume would have a larger impact on the estimate of the population mean. Therefore, it would be less likely for the volume of one individual bottle to fall within ±1 milliliter of μ compared to the sample mean of 16 bottles.

(b) To calculate the probability of the event described in part (a), we can use the z-score formula and the standard normal distribution table.

The given probability that x-bar estimates μ to within ±1 milliliter is 0.8176, which corresponds to a z-score of approximately 0.89 (based on the standard normal distribution table). Since we want to find the probability of the sample mean falling within ±1 milliliter of μ, we need to find the probability of the z-score being between -0.89 and 0.89 (i.e., within ±1 standard deviation from the mean).

Using the z-score formula: z = (x-bar - μ) / (σ / sqrt(n))

where μ = 298, σ = 3, n = 16 (sample size), and z = 0.89 (from the standard normal distribution table),

We can rearrange the formula to solve for x-bar:

0.89 = (x-bar - 298) / (3 / sqrt(16))

Simplifying:

0.89 = (x-bar - 298) / 0.75

Cross-multiplying:

x-bar - 298 = 0.89 * 0.75

x-bar - 298 = 0.6675

Adding 298 to both sides:

x-bar = 298 + 0.6675

x-bar ≈ 298.67

So, the probability of x-bar estimating μ to within ±1 milliliter is approximately 0.8176, which confirms our answer in part (a).

I hope it helped!

Step-by-step explanation:

The vertex is (50, 630). The vertex is the peak and correspond with the coordinates of he maximum height of the arch.

The solutions is the base of the arch and this (20, 0) and (80, 0)

vertex form, y = -0.7(x - 50)² + 630

Quadratic equation factored form, y = -0.7 (x - 80) (x - 20)

The equations are the same when plotted on a graph and examined mathematically. physically it looks different.

the domain is (-∞, ∞)

The height of the monument 15 feet from the left side is 472.5 feet

since the zeroes of the quadratic equation is (20, 0) and (80, 0), hence we have that

y = a(x - 20)(x - 80) and the equation pass points (50, 630)

630 = a(50 - 20)(50 - 80)

630 = a * 30 * -30

630 = -900a

a = -0.7

Quadratic equation has a standard vertex form, y = a(x - h)² + k    

y = a(x - h)² + k

vertex (h, k) = (50, 630) and a = -0.7

plugging the values

y = -0.7(x - 50)² + 630

Quadratic equation has a standard factored form, y = a(x - h)² + k    

y = a(x - r2)(x - r1)

where r2 and r1 are the roots r1 = 20 r2 = 80 an a = -0.7

plugging the values

y = -0.7 (x - 80)(x - 20)

The height of the monument 15 feet from the left side is gotten from the graph

this is 15 feet from 20 hence x = 35 feet

from the graph it can traced to be 472.5 feet

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Answer:

AB: y = 2x + 4

Step-by-step explanation:

Line AB is parallel to the line y = 2x + 3

When 2 lines are parallel they have the same coefficient of x.

=> AB: y = 2x + m (1)

Because line AB passes through the point (0, 4)

Replace x = 0; y = 4 into (1) => 2 × 0 + m = 4 => m = 4

So AB: y = 2x + 4

Answer:

...............................

Answer:

do 5x-27+7x-26=10x-23 because sum of two interior angle of a triangle is equal to the sum of one exterior angle and don't forget because it comes daily in exam

The statements that are correct concerning the outcome of events between Jasmine and her classmates include the following:

Most students found a pine tree.

More students found a pine tree than a birch tree. That is option A and D respectively.

The quantity of students that found birch tree = 1/7 = 0.14

The quantity of students that found pine tree = 7/9 = 0.8

The quantity of students that found maple tree = 1/4 = 0.25

The quantity of students that found oak tree = 11/23 = 0.48

Therefore, the statement that are correct about the outcome of the event between Jasmine and her classmates is as follows:

Most students found a pine tree.

More students found a pine tree than a birch tree.

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Equation of circle in standard form- (x + 1)² + (y + 2)² = 1².  radius of the circle is found as 1. centre -(h,k) =  (-1, -2)

We can solve quadratic equations that lack a factor by completing square.

In order to make the left side of an equation a perfect square trinomial, the equation's form must be adjusted.

Given equation is-

x² + y² + 2x + 4y + 4 = 0

Add and subtract 1 in the equation:

x² + y² + 2x + 4y + 4 + 1 - 1 = 0

rearranging:

(x² + 2x + 1) + (y² + 4y + 4) - 1 = 0

This can be written as:

(x² + 2*1x + 1²) + (y² + 2*2y + 2²) = 1

(x + 1)² + (y + 2)² = 1²  ...eq 1

Thus, the standard form of the equation of circle is:

(x - h)² + (y - k)² = r²  ..eq 2

On comparing eq 1 and eq 2

In which radius of the circle is found as 1.

centre -(h,k) =  (-1, -2)

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The number of people who play only football is |A' ∩ B' ∩ C'| = 25. So, 25 people play only football.

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It involves the use of various methods and techniques to draw conclusions from data and make informed decisions.

The main goal of statistics is to provide a systematic approach to understanding and interpreting data, which can be used in a wide variety of fields, including business, social sciences, engineering, medicine, and many others. Statistics is used to study and analyze various types of data, including numerical, categorical, and ordinal data, as well as time series and spatial data.

Let A, B, and C be the sets of people who play football, baseball, and hockey, respectively. We know that:

|A| = 61, |B| = 65, |C| = 72

We also know that:

|A ∩ B ∩ C| = 22, |A ∪ B ∪ C| = 141, |A' ∩ B' ∩ C'| = 11

where A', B', and C' denote the complements of A, B, and C, respectively.

We can use the principle of inclusion-exclusion to find the number of people who play only two games. This principle states that:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

Using the given values, we can substitute and simplify to get:

141 = 61 + 65 + 72 - |A ∩ B| - |A ∩ C| - |B ∩ C| + 22

|A ∩ B| + |A ∩ C| + |B ∩ C| = 75

We also know that an equal number of people play only two games, so let x be that number. Then:

|A ∩ B| = |A ∩ C| = |B ∩ C| = x

Substituting into the previous equation, we get:

3x = 75

x = 25

Therefore, 25 people play only two games. To find the number of people who play only football, we need to subtract the number of people who play baseball and hockey from the number of people who play only two games:

|A' ∩ B ∩ C| = x = 25

|A' ∩ B ∩ C'| = 11

|A' ∩ C ∩ B'| = x = 25

|A ∩ B' ∩ C'| = x = 25

|A' ∩ B| = 65 - (25 + 11) = 29

|A' ∩ C| = 72 - (25 + 11) = 36

|A' ∩ B' ∩ C| = 61 - (25 + 11) = 25

|A ∩ B' ∩ C| = 141 - (29 + 25 + 36 + 11) = 40

Therefore, the number of people who play only football is:

|A' ∩ B' ∩ C'| = 25

So, 25 people play only football.

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None of the given options A, B, C, or D is correct as they all provide different answers.

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

The question asks for the equivalent of 6 × 2. This means that we need to find a number that is equal to the result of multiplying 6 and 2 together.

When we multiply 6 and 2, we get:

6 × 2 = 12

So, the equivalent of 6 × 2 is 12.

However, none of the answer options provided matches this answer.

Option A suggests that the equivalent of 6 × 2 is 2 × 1, which is equal to 2, not 12.

Option B suggests that the equivalent of 6 × 2 is 3 × 2, which is equal to 6, not 12.

Option C suggests that the equivalent of 6 × 2 is 9 × 3, which is equal to 27, not 12.

Option D suggests that the equivalent of 6 × 2 is 18 × 1/2, which is equal to 9, not 12.

Therefore, none of the options provided is correct.

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Answer:

122.30 cm²

Step-by-step explanation:

Divide the polyhedron into shapes:

+) 2 triangles with the same area.

The area of the triangle is
(4.3×11)÷2 = 23.65 cm².
And with two triangles of the same area we take the sum of both areas
23.65 + 23.65 = 47.3 cm²

+) 3 rectangles with different areas.

(3×6) + (3×8) + (3×11) = 75 cm²

So the surface area is the sum of areas of the triangles and rectangles

47.3 + 75 = 122.3 = 122.30 cm²

Answer:

They are collinear if they are on the same line

By assuming that the farmer has goat and cows in the ratio 3:4, the  number of cows in the farm will be 32 cows.

To find out how many cows are in the farm, we need to know the total number of animals in the farm. Assuming the ratio of goats to cows is 3:4, we can write this as: [tex]3x : 4x[/tex]

Where 3x represents the number of goats, and 4x represents the number of cows. If we know that there are 24 goats, we can set up an equation to solve for x:

3x = 24

Dividing both sides by 3, we get:

x = 8

Now that we know the value of x, we can find the number of cows:

= 4x

= 4(8)

= 32

Therefore, there are 32 cows in the farm.

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The line's equation in point-slope form is shown here. Point (2, 5) is the given point on the line, and slope 2 is the given slope of the line. The slope of this line is -1/2, which is the negative reciprocal of the slope.

A line's point slope form equation is [tex]y - y_1 = m(x - x_1)[/tex]. Consequently, y - 0 = m(x = 0), or y = mx, is the equation of a line passing through the origin with a slope of m.

We require a point on the line and the slope of the line in order to create a line equation in point-slope form. In point-slope form,

[tex]y - y1 = m(x - x1)[/tex]

As an illustration, suppose we want to formulate the equation of the line passing through the coordinates (2, 5) and having a slope of 2. The values can be entered into the point-slope form as follows:

y - 5 = 2(x - 2)Let's say the given line has the equation [tex]y - y1 = m(x - x1)[/tex], where (x1, y1) is a point on the line and m is the slope of the line.

we can use the given point (2, 5). Then we can plug in the values into the point-slope form:

[tex]y - 5 = (-1/2)(x - 2).[/tex]

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The height of the cylinderical can is approximately 1147.8 inches.

What is a cylinder?

A cylinder is a three-dimensional geometric object made up of two parallel and congruent circular bases that are joined by a curving lateral surface. The lateral surface has a uniform circular cross-section and is perpendicular to the bases. A cylinder's height is the perpendicular distance between its bases. Cans, tubes, and pipes are all instances of cylinders. The volume of a cylinder may be computed by multiplying its base area by its height, and the formula for the lateral surface area is 2πrh, where r is the radius of the base and h is the cylinder's height.

Now,

We can start by finding the radius of the can using the given diameter:

radius = diameter/2 = 5/2 = 2.5 inches

The surface area of the can is given by:

Surface area = 2πrh + 2πr²

where h is the height of the can, r is the radius, and π is the mathematical constant pi.

Substituting the given values, we get:

18064 = 2π(2.5)(h) + 2π(2.5)²

Simplifying and solving for h, we get:

18064 = 15.707h + 39.269

18064 - 39.269 = 15.707h

18024.731 = 15.707h

h = 18024.731/15.707

h ≈ 1147.8 inches

Therefore, the height of the can is approximately 1147.8 inches.

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Answer: C, 125

Step-by-step explanation: That is the slope of the line, which remains constant. The slope represents the distance over the time, and distance divided by time equals the speed. This means that the speed remains constant throughout.

Answer:

a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:

r^2 - 2r - 3 = 0

Factoring, we get:

(r - 3)(r + 1) = 0

So the roots are r = 3 and r = -1.

The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:

y_h = c1e^3x + c2e^(-x)

To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = Ae^4x

Taking the first and second derivatives of y_p, we get:

y_p' = 4Ae^4x

y_p'' = 16Ae^4x

Substituting these into the original differential equation, we get:

16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x

Simplifying, we get:

5Ae^4x = e^4x

So:

A = 1/5

Therefore, the particular solution is:

y_p = (1/5)*e^4x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^3x + c2e^(-x) + (1/5)*e^4x

b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:

r^2 + r - 2 = 0

Factoring, we get:

(r + 2)(r - 1) = 0

So the roots are r = -2 and r = 1.

The general solution to the homogeneous equation y'' + y' - 2y = 0 is:

y_h = c1e^(-2x) + c2e^x

To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax + B)e^x

Taking the first and second derivatives of y_p, we get:

y_p' = Ae^x + (Ax + B)e^x

y_p'' = 2Ae^x + (Ax + B)e^x

Substituting these into the original differential equation, we get:

2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x

Simplifying, we get:

3Ae^x = 3xe^x

So:

A = 1

Therefore, the particular solution is:

y_p = (x + B)e^x

Taking the derivative of y_p, we get:

y_p' = (x + 2 + B)e^x

Substituting back into the original differential equation, we get:

(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x

Simplifying, we get:

-xe^x - Be^x = 0

So:

B = -x

Therefore, the particular solution is:

y_p = xe^x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^(-2x) + c2e^x + xe^x

c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:

r^2 - 9r + 20 = 0

Factoring, we get:

(r - 5)(r - 4) = 0

So the roots are r = 5 and r = 4.

The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:

y_h = c1e^4x + c2e^5x

To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax^2 + Bx + C)e^4x

Taking the first and second derivatives of y_p, we get:

y_p' = (2Ax + B)e^4x + 4Axe^4x

y_p'' = 2Ae^4x +

The equations that can be used to solve for y are y(y + 5) = 750,  y² – 5y = 750 and y(y – 5) + 750 = 0

What is quadratic equation?

it's a second-degree quadratic equation which is an algebraic equation in x. Ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term, is the quadratic equation in its standard form.

The equations that can be used to solve for y, the length of the room are:

1. y(y + 5) = 750

2. y² – 5y = 750

3. y(y – 5) + 750 = 0

Option 1 and 2 are quadratic equations that can be solved by factoring, completing the square, or using the quadratic formula. Option 3 is also a quadratic equation but it requires rearranging to the standard form before applying the same methods. The last option, (y + 25)(y – 30) = 0, is not a quadratic equation but it can be easily solved using the zero product property.

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Answer:

Option A

Step-by-step explanation:

Solving using quadratic formula:

Use the quadratic formula to determine the coefficient of the quadratic equation.

Option A:

         x² + 6x + 24 = 3x

        x² + 6x + 24 - 3x = 0

         x² + 3x + 24 = 0

Compare with ax² + bx + c = 0,

a = 1 ; b = 3 and c = 24

[tex]\boxed{\bf x= \dfrac{-b \± \sqrt{b^2-4ac}}{2a}}[/tex]

     [tex]= \dfrac{-3 \±\sqrt{3^2-4(1)(24)}}{2(1)}[/tex]

This equation could be solved using the application of the quadratic formula.

Option B:

3x² + 3x + 24 = 3

3x² + 3x + 24 - 3 = 0

3x² + 3x + 21 = 0

a = 3  ; b = 3 and c = 21

Here, co-efficients are different.

Option C:

     x² + 3x - 24 = 3

 x² + 3x - 24 - 3= 0

   x² + 3x - 27    = 0

          a = 1  ; b = 3 and c = -27

Here, co-efficients are different.

Option C:

     x² + 3x + 24 = 3

 x² + 3x + 24 - 3 = 0

    x² + 3x + 21    = 0

             a= 1 ; b = 3  and c = 21

Here, co-efficients are different.

Answer:

0

Step-by-step explanation:

0

The simplified expression is adbc - bc - abd√c - b√ad√c + acd√b + cd√ab.

In mathematics, the distributive property is a property of operations that involves multiplying a factor by a sum or difference of terms. The distributive property states that the product of a factor and a sum or difference of terms is equal to the sum or difference of the products of the factor with each term individually. In other words, for any numbers a, b, and c, the distributive property can be expressed as:

a(b + c) = ab + ac

Given expression

(a√b+c√d)-b√a)×(d√c-√ab+abc)

Simplifying the expression

(a√b+c√d-b√a)×(d√c-√ab+abc)

by simplifying the given expression using the distributive property of multiplication:

(a√b+c√d-b√a)×(d√c-√ab+abc)

= adbc + acd√b - abd√c - b√ad√c + cd√ab - bc

= adbc - bc - abd√c - b√ad√c + acd√b + cd√ab

Therefore, the simplified expression is adbc - bc - abd√c - b√ad√c + acd√b + cd√ab.

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A. There are 9 sample spaces.

B.  The probability of choosing Chocolate, Fudge, and Rainbow is 1/9

What is meant by sample spaces?

In probability theory, a sample space is the set of all possible outcomes of a random experiment or process. It is used to define the space of events and calculate probabilities.

What is meant by probability?

Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. It is calculated by dividing the number of favourable outcomes by the total number of possible outcomes.

According to the given information

A. There are 9 possible combinations (3 ice cream flavors x 3 toppings x 1 sprinkle color), so there are 9 sample spaces.

B.  The probability of choosing Chocolate, Fudge, and Rainbow is 1/9, assuming all combinations are equally likely.

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Answer: 4

Step-by-step explanation:

To find the unit rate, we can cross-multiply the fractions.

Multiplying the numerator of the first fraction by the denominator of the second fraction, we get 1/5.

Multiplying the numerator of the second fraction by the denominator of the first fraction, we get 1/20.

Now we have the equation 1/5 = 1/20.

To solve for the unknown, we can cross-multiply again, which gives us 20 * 1/5 = 4.

Therefore, the unit rate is 4.

Answer:

a) To find out how big of a loan you can afford, we can use the formula for the monthly payment of a mortgage:

M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ]

where M is the monthly payment, P is the principal (the amount borrowed), i is the monthly interest rate (which is the annual interest rate divided by 12), and n is the number of monthly payments (which is the number of years times 12).

In this case, we know that M = $1,000, i = 0.053/12, and n = 30 x 12 = 360. We want to solve for P, the principal we can afford.

Substituting these values into the formula, we get:

$1,000 = P [ 0.004416(1 + 0.004416)^360 ] / [ (1 + 0.004416)^360 - 1 ]

Simplifying and solving for P, we get:

P = $183,928.72

Therefore, you can afford a loan of approximately $183,928.72.

b) The total money paid to the loan company will be the monthly payment multiplied by the number of payments over the life of the loan. In this case, we have:

Total money paid = $1,000 x 360 = $360,000

Therefore, the total amount of money paid to the loan company will be $360,000.

c) To find out how much of that money is interest, we can subtract the principal from the total amount paid. In this case, we have:

Interest paid = Total money paid - Principal = $360,000 - $183,928.72 = $176,071.28

Therefore, the amount of money paid in interest will be $176,071.28.

There are 7 books in the stack.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.com

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

To find the number of books in the stack, we can divide the total mass of the stack by the mass of each book:

Number of books = Total mass of the stack / Mass of each book

In this case, the total mass of the stack is 21 kilograms, and the mass of each book is 3 kilograms.

Number of books = 21 kg / 3 kg/book

Simplifying the right side, we get:

Number of books = 7 books

Therefore,

There are 7 books in the stack.

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For the given expression a)(f◦g)(4) = 1. b)(g◦f)(2) = -29. c)(f◦f)(1) = 1. d)(g◦g)(0) = -17.5.

An expression refers to a combination of numbers, symbols, and/or operators that represents a mathematical value or relationship. It can be a single term, such as a number or a variable, or a combination of terms connected by operators, such as addition, subtraction, multiplication, division, exponentiation, etc.

A function is a relation between a set of inputs (the domain) and a set of possible outputs (the codomain), such that each input is associated with exactly one output.

According to the given information:

(a) To find (f◦g)(4), we need to first substitute g(4) into f(x) in the given expression, and then evaluate f(g(4)).

Given:

f(x) = 3x² - 2

g(x) = 7 - (1/2)x²

Substituting g(4) into f(x):

g(4) = 7 - (1/2)(4²) = 7 - 8 = -1

Now, substituting -1 into f(x):

f(-1) = 3(-1)² - 2 = 3 - 2 = 1

So, (f◦g)(4) = 1.

(b) To find (g◦f)(2), we need to first substitute f(2) into g(x), and then evaluate g(f(2)).

Substituting f(2) into g(x):

f(2) = 3(2)² - 2 = 12

Now, substituting 12 into g(x):

g(12) = 7 - (1/2)(12)^2 = 7 - 36 = -29

So, (g◦f)(2) = -29.

(c) To find function (f◦f)(1), we need to first substitute f(1) into f(x), and then evaluate f(f(1)).

Substituting f(1) into f(x):

f(1) = 3(1)²- 2 = 1

Now, substituting 1 into f(x):

f(1) = 3(1)² - 2 = 1

So, (f◦f)(1) = 1.

(d) To find (g◦g)(0), we need to first substitute g(0) into g(x), and then evaluate g(g(0)).

Substituting g(0) into g(x):

g(0) = 7 - (1/2)(0)² = 7

Now, substituting 7 into g(x):

g(7) = 7 - (1/2)(7)² = 7 - 24.5 = -17.5

So, (g◦g)(0) = -17.5.

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With given compound interest, it will take 27.25 years to pay off the loan.

What is Compound interest?

Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, the interest that is earned on both the principal amount and any previously accumulated interest. In other words, it is interest that is calculated not only on the original amount of money but also on any interest that has been earned previously. This can result in significant growth of an investment or loan balance over time, as the interest earned on the accumulated interest can compound exponentially. Compound interest can be calculated using a specific formula that takes into account the principal amount, the interest rate, and the compounding frequency.

Now,

First, we need to calculate the total amount of the loan. We round up the tuition and fees cost for 5 years to the nearest thousand dollars:

Total cost = $70,000

Rounded up to nearest thousand = $70,000

So, the amount of the loan is $70,000.

Next, we can use the formula N= (-log⁡(1-i*A/P))/(log⁡(1+i)) to calculate the number of monthly payments needed to pay off the loan.

N = (-log(1-0.0025*70000/250))/(log(1+0.0025))

N = (-log(1.75))/(log(1.0025))

N = 326.45

Rounding up to the nearest whole number, it will take 327 monthly payments to pay off the loan.

So, it will take 327/12 = 27.25 years to pay off the loan.

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