due soon!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer:

7)a. Brad--2/3, Lola--1/3

b. 2x + x = $630

3x = $630, so x = $210

Brad--$420, Lola--$210

8)a. William--3/7, Emma--4/7

b. 3x + 4x = $12,600

7x = $12,600, so x = $1,800

William--$5,400, Emma--$7,200

There were 60 adults, 318 teenagers, and 192 preteens at the concert.

Let's use the variables "A", "T", and "P" to represent the number of adults, teenagers, and preteens, respectively, who attended the concert. We can start by using the information given in the problem to set up a system of equations:

A + T + P = 570 (equation 1)

5A + 3T + 2P = 1950 (equation 2)

We also know that "three-fourths as many teenagers as preteens attended," which we can write as:

T = (3/4)P (equation 3)

Now we can substitute equation 3 into equation 1 to eliminate T:

A + (3/4)P + P = 570

Simplifying:

A + (7/4)P = 570

Multiplying both sides by 4 to eliminate the fraction:

4A + 7P = 2280 (equation 4)

We can now use equations 2 and 4 to solve for A and P. Multiplying equation 4 by 5 and subtracting it from equation 2 (to eliminate A) gives:

5A + 3T + 2P = 1950

-(20A + 35P = 11400)

-17A - 32P = -9450

Simplifying:

17A + 32P = 9450 (equation 5)

Now we have two equations with two variables (equations 4 and 5), which we can solve using substitution or elimination. We'll use substitution:

From equation 4, we can solve for A in terms of P:

4A + 7P = 2280

4A = 2280 - 7P

A = (2280 - 7P)/4

Substituting this into equation 5, we get:

17[(2280 - 7P)/4] + 32P = 9450

Simplifying:

3915 - 119P + 32P = 9450

13P = 5535

P = 425

Now we can use equation 1 to solve for T:

A + T + P = 570

A + (3/4)P + P = 570

A + (7/4)P = 570

A + (7/4)(425) = 570

A = 60

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(a) To find the probability that a student is both a senior and an English major, we need to use the formula:
P(A and B) = P(A) x P(B|A)
where A represents the event of being a senior and B represents the event of being an English major.

We know that there are seniors and English majors in the class, but we don't know how many seniors are English majors. Therefore, we cannot use the formula directly. However, we do know that students are neither seniors nor English majors.

Let's use a Venn diagram to represent this information:

[Insert Venn diagram]

The total number of students in the class is the sum of the three regions:
Total = Seniors + English majors + Neither
= + +

But we are not given any of these values. However, we do know that the number of students who are neither seniors nor English majors is . Therefore:

Total = Seniors + English majors + Neither
= + +
=

Now we can find the probability that a student is both a senior and an English major:

P(Senior and English major) = P(A and B) =

(b) Given that the selected student is a senior, we only need to consider the seniors region of the Venn diagram:

[Insert Venn diagram with only seniors]

We know that students are seniors, but we don't know how many of them are also English majors. Let's call this number X:

[Insert Venn diagram with X seniors who are also English majors]

The probability that a senior student is also an English major is given by:

P(English major|Senior) = X /

We can find X by using the fact that students are neither seniors nor English majors:

Total = Seniors + English majors + Neither
= + +
=

Since we know that there are seniors and that students are neither seniors nor English majors, we can conclude that:

Total = Seniors + Neither
= +
=

Solving for Neither, we get:

Neither =

Now we can find X:

X = Seniors - Neither
= -
=

Plugging this value into the formula for conditional probability, we get:

P(English major|Senior) = X /
= /
=

Therefore, the probability that a senior student is also an English major is .

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The obtuse angle between the hands of the clock at 9:12 is 285 degrees.

To find the angle between the hands of a clock at any given time, we use the formula:

angle = |(30H - 11/2 M)|

Here H is the hour hand position (counted from 12 o'clock) and M is the minute hand position (counted from 12 o'clock).

At 9:12, the hour hand is slightly past the 9 and the minute hand is at the 12.

The hour hand has moved 9/12 of the way from 9 to 10, which is 9/12 * 30 = 22.5 degrees.

The minute hand is at 12, which corresponds to 0 degrees.

Substituting H = 9.5 and M = 0 into the formula above, we get:

angle = |(30(9.5) - 11/2(0))| = 285 degrees

Thus, the obtuse angle between the hands of the clock at 9:12 is 285 degrees.

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The amount of material needed = the surface area of the cylindrical can which is approximately calculated as: 332 square centimeters.

The material used = surface area of cylindrical can = 2πr(h + r).

Given the dimensions of the cylindrical can, we have:

radius (r) = 7/2 = 3.5 cm

height of cylindrical can (h) = 11.6 cm

π = 3.14

Amount of tin used = surface area = 2 * 3.14 * 3.5 * (11.6 + 3.5)

≈ 332 square centimeters

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In Dilation the transformation could be used to show that triangle Q is similar but not congruent to triangle P.

Transformations refer to a set of techniques used to alter the shape and position of a point, line, or geometric figure. Dilation is one of the basic operations in mathematical morphology and it is usually represented by ⊕. When performing a dilation operation, a structuring element is typically utilized to examine and enlarge the shapes present in the input image.

The angle measures would be the same, and the ratio of corresponding sides would be equal to the scale factor used in the dilation. The transformation could be used to show that triangle Q is similar but not congruent to triangle P in Dilation.

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The volume of the solid of revolution is (31/30)π.

To find the volume of the solid of revolution, we need to use the method of cylindrical shells. We will integrate over the height of the region, which is from y=0 to y=1.

First, let's find the points of intersection between the curves:

[tex]x - x^3\\ = x - x^2x^3 - x^2\\ = 0x^2(x-1) \\= 0x=0, x=1[/tex]

So the region we need to rotate is bound by the curves x=0, x=1, y=x-x^3 and y=x-x^2.

Next, we need to express the curves in terms of x and y as follows:

[tex]x = y + y^3\\x = y + y^2[/tex]

To use the method of cylindrical shells, we need to express the radius of each shell as a function of y. The radius of each shell is the distance from the y-axis to the curve at a given height y.

The distance from the y-axis to the curve [tex]x = y + y^3[/tex] is simply [tex]x = y + y^3.[/tex]Therefore, the radius of each shell is r = y + y^3.

The distance from the y-axis to the curve [tex]x = y + y^2 is x = y + y^2.[/tex]Therefore, the radius of each shell is[tex]r = y + y^2.[/tex]

The volume of each shell is given by the formula V = 2πrhΔy, where h is the height of the shell (which is simply Δy) and Δy is the thickness of each shell.

Thus, the total volume of the solid of revolution is given by the integral:

[tex]V = ∫[0,1] 2π(y+y^3)(y+y^2) dy\\V = 2π ∫[0,1] (y^4 + 2y^3 + y^2) dy\\V = 2π [(1/5)y^5 + (1/2)y^4 + (1/3)y^3] [0,1]\\V = 2π [(1/5) + (1/2) + (1/3)]V = (31/30)π[/tex]

Therefore, the volume of the solid of revolution is (31/30)π.

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The given identity sin(0) + sin(o) = 2 sin(2*0 - o) * cos(o) is false when checked using complex exponentials and Euler's formula.

To check the given identity using complex exponentials, we'll make use of Euler's formula, which states:

e^(ix) = cos(x) + i*sin(x)

Let's rewrite the given identity in terms of complex exponentials:

sin(0) + sin(o) = 2 sin(2*0 - o) * cos(o)

Now, we'll apply Euler's formula:

(1/2i)(e^(i0) - e^(-i0)) + (1/2i)(e^(io) - e^(-io)) = 2(1/2i)(e^(i(2*0 - o)) - e^(-i(2*0 - o))) * (1/2)(e^(io) + e^(-io))

Simplify the expression:

(1/2i)(e^(i0) - e^(-i0) + e^(io) - e^(-io)) = (1/2i)(e^(i(2*0 - o)) - e^(-i(2*0 - o))) * (1/2)(e^(io) + e^(-io))

We notice that the left side of the equation does not match the right side, which means that the given identity is not true. Therefore, the answer is:

2. FALSE

The given identity sin(0) + sin(o) = 2 sin(2*0 - o) * cos(o) is false when checked using complex exponentials and Euler's formula.

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The triangle is obtuse, since the square of the longest side is greater than the sum of the squares of the other two sides.

The sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse:

a²+b²=c²

a, b and c are side lengths

a=12,b=37 and c=40

12²+37²=40²

144+1369=1600

1513 is not equal to 1600

Since 1513 < 1600, we know that:

This means that the triangle is obtuse, since the square of the longest side is greater than the sum of the squares of the other two sides.

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If Meghan has y dollars and Joseph has half as much money as Meghan then the expression y ÷ 2 represents the amount of money Joseph has. Thus, the most appropriate option which is the answer to the given question is B.

An expression is defined as a mathematical phrase with two or more variables with any of the mathematical operations between them. The following are a few examples of expressions: 3x + 45y, 9u, 55 - a, and so on.

In the given question, We are given,

Amount of money Meghan has = $y

Amount of money Joseph has = half as much as Meghan

Thus to calculate the money owned by Joseph, we have to divide the money of Meghan by 2.

And we get the amount of money Joseph has = y ÷ 2

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The complete question might have been:

Meghan has y dollars. Joseph has half as much money as Meghan. Which expression represents the amount of money Joseph has?

A. y + 2

B. y * 2

C. y ÷ 2

D. y - 2

The area of the irregular shape for this problem is given as follows:

A = 60 ft².

The area of the composite figure is given by the sum of the areas of all the parts that compose the figure.

The shape in this problem is composed by three rectangles, with dimensions given as follows:

Hence the area of the shape is obtained as follows:

A = 4 x 6 + 4 x 5 + 4 x 4

A = 60 ft².

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The probability that the paint thickness at a particular point is less than 0.2mm is approximately 0.14.

a) To find the value of A, we need to use the fact that the total area under the probability density function must be equal to 1:

∫0.5 0.125 f(x) dx = 1

Using the given formula for f(x), we get:

∫0.5 0.125 A(0.5 - (x - 0.25)2) dx = 1

Expanding the square inside the integral, we get:

∫0.5 0.125 A(0.5 - x2 + 0.5x - 0.0625) dx = 1

Simplifying and integrating, we get:

A(0.5x - 1/3 x3 + 0.5(1/2)x2 - 0.0625x)∣∣0.125^0.5 = 1

Substituting the limits of integration and simplifying, we get:

A(1/48) = 1

Therefore, A = 48.

The probability density function can be sketched by plotting the function f(x) against x for values of x between 0.125 and 0.5. It will look like a bell-shaped curve with its maximum value at x = 0.25 and decreasing to 0 at x = 0.125 and x = 0.5.

b) The cumulative distribution function (CDF) is defined as:

F(x) = P(X ≤ x) = ∫(-∞)x f(t) dt

To construct the CDF, we need to integrate the probability density function from 0.125 to x:

F(x) = ∫x 0.125 f(t) dt

Using the formula for f(x), we get:

F(x) = 48∫x 0.125 (0.5 - (t - 0.25)2) dt

Simplifying the integral and integrating, we get:

F(x) = 16(x - 0.25) + 3/2(x - 0.25)3 - 1/16(x - 0.25)4

The cumulative distribution function can be sketched by plotting F(x) against x for values of x between 0.125 and 0.5. It will start at 0 when x = 0.125 and approach 1 when x = 0.5, and will be an increasing curve with a maximum slope at x = 0.25.

c) To find the probability that the paint thickness at a particular point is less than 0.2mm, we need to evaluate the cumulative distribution function at x = 0.2:

P(X ≤ 0.2) = F(0.2) = 16(0.2 - 0.25) + 3/2(0.2 - 0.25)3 - 1/16(0.2 - 0.25)4

P(X ≤ 0.2) ≈ 0.14

Therefore, the probability that the paint thickness at a particular point is less than 0.2mm is approximately 0.14.

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-12x will be the value of the linear portion in this quadratic equation. Thus, option A is correct.

A linear portion will establish a condition that the value should have the power of the variable as 1. j

In the given equation 7[tex]x^{2}[/tex] - 12x + 16 = 0 which is a trinomial equation:

7[tex]x^{2}[/tex] will have a power of 2

- 12x have a power of 1

16 has a power of o.

The condition of the linear equation states that the value of power should be equal to one. Therefore, option A is correct.

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The question is incomplete, Complete question probably will be  is:

Select the term that describes the linear portion in this quadratic equation.

7[tex]x^{2}[/tex] - 12x + 16 = 0

a. -12x

b. 7x2

c. 16

To determine the coordinates of vector AT in this base, we first need to find the coordinates of points B' and C.

Let's start by finding the coordinates of B'. Since B' divides side CA in the ratio 4:4 from C, we can use the following formula to find its coordinates:

B' = (4C + 4A)/8

Simplifying this expression, we get:

B' = (C + A)/2

Similarly, we can find the coordinates of C using the fact that it divides AB in the ratio 3:5 from A:

C = (3A + 5B)/8

Now, we can use these coordinates to find the equation of lines BB' and CC.

The equation of line BB' can be found using the point-slope form:

BB': y - yB = (yB' - yB)/(xB' - xB) * (x - xB)

Substituting the coordinates of B and B', we get:

BB': y - 0 = (yB'/2)/(xB'/2) * (x - 1)

Simplifying this expression, we get:

BB': y = (yB'/xB') * x - yB'/2

Similarly, we can find the equation of line CC:

CC: y = (yC'/xC') * x - yC'/2

Now, we can find the coordinates of point T by solving the system of equations formed by the equations of lines BB' and CC:

(yB'/xB') * x - yB'/2 = (yC'/xC') * x - yC'/2

Simplifying this expression, we get:

x = (yC' - yB') / ((yC'/xC') - (yB'/xB'))

Substituting this value of x into the equation of line BB', we get:

y = (yB'/xB') * ((yC' - yB') / ((yC'/xC') - (yB'/xB'))) - yB'/2

Simplifying this expression, we get:

y = ((yB' * xC') - (yC' * xB')) / (2 * (xC' - xB'))

Now, we can find the coordinates of point T:

T = (x, y)

Substituting the coordinates of B', C, and T into the expression for vector AT, we get:

AT = T - A

Simplifying this expression, we get:

AT = ((x - xA), (y - yA))
In triangle ABC, let B' and C' be points on sides CA and AB, respectively. B' divides CA in a 4:4 ratio from C, meaning CB':B'A = 4:4, and C' divides AB in a 3:5 ratio from A, meaning AC':C'B = 3:5.

Let A be the origin, and let AB = a and AC = b be the vectors forming a base in the plane. Since B' divides CA in half, the position vector of B' is the midpoint of CA, so B' = (1/2)b. Similarly, C' divides AB in a 3:5 ratio, so C' = (5/8)a.

Now, let's consider triangle B'C'T. Since T is the intersection of BB' and CC', we can write the vectors BT and CT in terms of B'T and C'T, respectively:

BT = B'T + TB
CT = C'T + TC

Since B'T and C'T are parallel to b and a, respectively, we can write:

BT = k1 * b
CT = k2 * a

Here, k1 and k2 are constants. Now, using the ratios CB':B'A and AC':C'B, we can write:

k1 * b = 4 * (TB - (1/2)b)
k2 * a = 5 * (TC - (5/8)a)

Finally, we want to find the vector AT. Since AT = TB + TC, we can substitute the expressions for TB and TC from the above equations and solve for AT:

AT = (k1 * b + (1/2)b) + (k2 * a + (5/8)a)

The coordinates of the vector AT are given in terms of the base vectors a and b, with the constants k1 and k2 accounting for the position of the point T in the triangle.

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A. To find a linear function, we need to determine the slope and y-intercept of the equation. Using the two data points given:
Slope = (50-20)/($75-$120) = 30/(-45) = -2/3
Using point-slope form, we can find the equation:
y - 50 = (-2/3)(x - 75)
y = (-2/3)x + 100
Therefore, the linear function is: p = (-2/3)x + 100, where x is the weekly sales and p is the price in dollars.
B. The suitable applied domain for this function is the range of weekly sales that the retailer expects to encounter. In this case, it could be any value greater than 0 and less than or equal to 50 (since that was the highest weekly sales number given in the data).
C. To find the price the retailer should set to sell 80 PortaBoys next week, we can plug in x = 80 into the linear function:
p = (-2/3)(80) + 100
p = $46.67
Therefore, the retailer should set the price at $46.67 to sell 80 PortaBoys next week.

A. To find the linear function, we'll use the given data points: (x1, p1) = (20, 120) and (x2, p2) = (50, 75). The general form of a linear function is p = m*x + b, where m is the slope and b is the y-intercept.

First, we find the slope (m):
m = (p2 - p1) / (x2 - x1)
m = (75 - 120) / (50 - 20)
m = (-45) / 30
m = -1.5

Now, we'll use one of the data points to find the y-intercept (b). Let's use (20, 120):
120 = -1.5 * 20 + b
120 = -30 + b
b = 150

So, the linear function fitting this data is: p(x) = -1.5x + 150

B. A suitable applied domain for this function would be x ∈ [0, ∞), meaning the number of PortaBoy game systems sold should be greater than or equal to 0.

C. To find the price at which the retailer should sell the systems to sell 80 PortaBoys, we'll plug x = 80 into the function:
p(80) = -1.5 * 80 + 150
p(80) = -120 + 150
p(80) = 30

The retailer should set the price at $30 per system to sell 80 PortaBoys next week.

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The area of the rhombus given is 279 cm².

Given a rhombus.

Base length of the rhombus = 18 centimeters

Height of the rhombus = 15.5 centimeters

The formula to find the area of the rhombus is,

Area = Base × Height

        = 18 × 15.5

        = 279 cm²

Hence the required area of the rhombus is 279 cm².

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1. (a) The limit as (x,y) approaches (0,0) along this curve is 6/5.

(b) The limit as (x,y) approaches (0,0) along this curve is 3/5.

(c) The limit as (x,y) approaches (0,0) along this curve is 6.

2. The value of c that makes the function f(x,y) continuous at (0,0) is c=0.

1. (a) Along the curve y=2x1/2, we have y2 = 4x. Substituting this into the expression for f(x,y), we get:

f(x,y) = 6x/(4x + x) = 6x/5x = 6/5

Therefore, the limit as (x,y) approaches (0,0) along this curve is 6/5.

(b) Along the curve y=3x1/2, we have y2 = 9x. Substituting this into the expression for f(x,y), we get:

f(x,y) = 6x/(9x + x) = 6x/10x = 3/5

Therefore, the limit as (x,y) approaches (0,0) along this curve is 3/5.

(c) Along the curve y=mx1/2, we have y2 = m2x. Substituting this into the expression for f(x,y), we get:

f(x,y) = 6x/(m2x + x) = 6x/(m2 + 1)x

If m ≠ 0, then as x approaches 0, the denominator approaches m2 + 1, and the numerator approaches 0. Therefore, the limit is 0.

If m = 0, then the curve y=mx1/2 reduces to the x-axis. Along this curve, we have y = 0, so y2 = 0. Substituting this into the expression for f(x,y), we get:

f(x,y) = 6x/(0 + x) = 6

Therefore, the limit as (x,y) approaches (0,0) along this curve is 6.

Since the limit depends on the value of m, and there are different limits along different curves passing through (0,0), we conclude that the limit of f(x,y) does not exist as (x,y) approaches (0,0).

2. To make the function f(x,y) continuous at (0,0), we need to find the value of c such that the limit of f(x,y) as (x,y) approaches (0,0) along any path equals c. Specifically, we need the limit to exist and be equal to c along the x-axis and the y-axis.

Along the x-axis, we have y = 0, so f(x,0) = x. Therefore, the limit of f(x,y) as (x,y) approaches (0,0) along the x-axis is 0 + 0 + 2c = c.

Along the y-axis, we have x = 0, so f(0,y) = y4 + 2. Therefore, the limit of f(x,y) as (x,y) approaches (0,0) along the y-axis is 0 + 0 + c = c.

To make the function continuous at (0,0), we need both limits to be equal to c. Therefore, we must have:

c = c + 2c

2c = c

c = 0

Therefore, the value of c that makes the function f(x,y) continuous at (0,0) is c=0.

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A)

The equation for the number of fish after t years is

P(t) = 9400 (1 - 0.9574 x e^{-0.0899t})

B)

It will take approximately 7.31 years for the fish population to increase to 3000.

We have,

A)

The equation for logistic growth is:

dP/dt = rP(1 - P/K)

where P is the population size, t is the time in years, r is the intrinsic growth rate, and K is the carrying capacity.

We know that at t = 0, P = 400, and at t = 1, P = 570.

We can use this information to find the value of r:

570 - 400 = r(1 - 400/9400)

r ≈ 0.0899

Now we can use the logistic growth equation to find P(t):

dP/dt = 0.0899P(1 - P/9400)

∫dP/(P(1 - P/9400)) = ∫0.0899dt

-ln|1 - P/9400| = 0.0899t + C

|1 - P/9400| = e^(-0.0899t - C)

1 - P/9400 = ±e^(-0.0899t - C)

P = 9400(1 - Ce^(-0.0899t))

We can use the initial condition P(0) = 400 to find the value of C:

400 = 9400(1 - Ce^0)

C ≈ 0.9574

The equation for the number of fish after t years.

P(t) = 9400 (1 - 0.9574 e^{-0.0899t})

B)

To find how long it will take for the population to increase to 3000, we need to solve the equation P(t) = 3000 for t.

3000 = 9400(1 - 0.9574e^(-0.0899t))

Dividing both sides by 9400 and rearranging.

e^(-0.0899t) = 1 - 3000/9400

e^(-0.0899t) ≈ 0.6808

Taking the natural logarithm of both sides.

-0.0899t ≈ ln(0.6808)

t ≈ 7.31 years

It will take approximately 7.31 years for the fish population to increase to 3000.

Thus,

The equation for the number of fish after t years is

P(t) = 9400 (1 - 0.9574 e^{-0.0899t})

It will take approximately 7.31 years for the fish population to increase to 3000.

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As per the given graph, there are 8 fewer students walk to school in Class A than in Class B

In this case, we are looking at two classes, Class A and Class B, and the number of students who walk to school in each class. The graph should show a picture or symbol for each student who walks to school.

Now, to answer the question of how many fewer students walk to school in Class A than in Class B, we need to compare the number of symbols or pictures for each class on the graph.

One way to do this is to count the number of symbols or pictures for each class and then subtract the number of students who walk to school in Class A from the number of students who walk to school in Class B.

This will give us the number of students that walk to school in Class B but not in Class A, which is the same as the number of fewer students who walk to school in Class A.

If there are 10 symbols for Class A and 18 symbols for Class B, then we can say that there are 8 fewer students who walk to school in Class A than in Class B.

We get this by subtracting 10 from 18, which gives us 8.

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(0,8) is the y-intercept of the given function g(x).

The y-intercept of g from the equation for f(x).

However, to find the value of g(0) using the given equation:

g(0) = -4f(0) + 12

Since f(0) = 1, we can substitute:

g(0) = -4(1) + 12 = 8

So the value of g at x=0 is 8, and the y-intercept of the function g(x) will be 8.

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Complete question:

What is the y-intercept of function g if g ⁡ ( x ) = - 4 ⁢ f ⁡ ( x ) + 12 and f(0) = 1 ?

Answer:

[tex]7 {t}^{2} + 2t - 9 = [/tex]

[tex](7t + 9)(t - 1)[/tex]

Answer:The answer is C.

Step-by-step explanation:

The value of area of the window is, 554.65 inches².

We have to given that;

Kenia makes regular octagonal-shaped stained glass windows.

And, the apothem of the window is 14 inches.

We know that;

The area of an octagon is,

A = 2(1 + √2) a²,

where a represents the length of the apothem.

Plugging value of a = 14 inches, we get:

A = 2(1 + √2) 14²

A = 2(1.414) 196

A = 2.828  x 196

A = 554.656

Hence., After Rounding to the nearest tenth, the area of the octagonal-shaped stained glass window is approximately 554.7 square inches.

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This explanation is supported by the provided information and does not involve any cartographic errors or legislative changes affecting the electoral system.

In the 2008 election, Florida had 27 electoral votes. The data on this map can be explained by changes in the state's population and federal legislation affecting the electoral system. Population shifts, as revealed by census data, can lead to states gaining or losing representatives and electoral votes. In this case, if Florida experienced a significant population increase, it could result in additional representatives being allocated, thus increasing the number of electoral votes. On the other hand, changes in federal legislation can also impact the organization of the electoral system. However, there is no specific information provided about such legislative changes affecting Florida's electoral votes in this question. Therefore, the most plausible explanation for the data shown on the map is the population increase in Florida, leading to the state gaining representatives and electoral votes. In conclusion, the most likely reason behind the change in Florida's electoral votes is the increase in population, which resulted in additional representatives and electoral votes being allocated to the state.

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To ensure that the first child in line has the greatest number of candy after all 100 pieces are distributed, they must receive as many pieces as possible. We know that each child is guaranteed at least one piece of candy, which means that the 16 children will receive a minimum of 16 pieces altogether.

To find the maximum number of pieces the first child can receive, we can start by assuming that each of the remaining 15 children receive only one piece of candy. This would leave a total of 85 pieces for the first child to receive.

However, we want to maximize the number of pieces the first child can receive while still ensuring that each of the other children receives at least one piece. We can achieve this by giving the second child 2 pieces of candy, the third child 3 pieces, and so on, until we get to the 15th child who receives 15 pieces. This leaves a total of 40 pieces for the first child to receive, which is the maximum amount they can receive while still guaranteeing that each of the other children receives at least one piece.

Therefore, the first child must receive at least 40 pieces of candy to ensure that they have the greatest number after all 100 pieces are distributed.

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The equation for the average speed of the Kayaker's paddling would be 6(x + 1) + 6(x - 1) = 8 ( x^ 2 - 1).

The total time that the kayaker spent paddling is 2 hours and 40 minutes which can be written as :

2 + ( 40 / 60 )

= 2 + ( 2 / 3 )

= 8 / 3 hours

The equation for the average speed :

(2 miles ) / ( x - 1 mph ) + ( 2 miles ) / ( x + 1 mph ) = ( 8 / 3) hours

3 ( x - 1 ) ( x + 1) × ( 2 / ( x - 1 ) ) + 3 ( x  - 1 ) ( x + 1) × ( 2 / ( x  + 1 ) ) = ( 8 / 3) x 3 ( x  - 1 ) ( x  + 1 )

6(x + 1) + 6(x - 1) = 8 ( x ^2 - 1)

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The amount will become zero after 400 months.

Given that, Mr. Tallchief have $400,000 in his account after his retirement,

He intends to withdraw 4,000 from the account every month after the retirement,

We need to find the equation that represents the amount in the account n months after his retirement.

So,

f(n) = 400,000 - 4000n

This equation will give the withdrawal of money each month,

Now, when the account will have no money in it,

A = 0,

0 = 400,000 - 4000n

400,000 = 4000n

n = 400

Hence, the amount will become zero after 400 months.

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The correct answer is A. f(2,1) = -8, relative minimum and B. f(1,2) = 9, relative minimum.

To find relative extrema of the function f(x,y) = x³ - 12xy + 8y³, we first find the partial derivatives f_x and f_y:

f_x = 3x² - 12y
f_y = -12x + 24y²

Set both partial derivatives equal to 0 to find critical points:

3x² - 12y = 0 => x² = 4y
-12x + 24y² = 0 => x = 2y²

Solving these equations simultaneously, we get the critical points (2,1) and (1,2). To determine if these points are relative minima or maxima, we use the second derivative test. Compute the second partial derivatives:

f_xx = 6x
f_yy = 48y
f_xy = f_yx = -12

Evaluate the discriminant D = (f_xx * f_yy) - (f_xy * f_yx) at each critical point:

D(2,1) = (12 * 48) - (-12 * -12) = 576 - 144 = 432 > 0, and f_xx(2,1) = 12 > 0, so it's a relative minimum with value f(2,1) = -8.

D(1,2) = (6 * 96) - (-12 * -12) = 576 - 144 = 432 > 0, and f_xx(1,2) = 6 > 0, so it's a relative minimum with value f(1,2) = 9.

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The derivative is a. y' = [7 * cos(7x) * tan(3x) - 3 * sin(7x) * sec²(3x)] / [tan²(3x)] and the derivative of second funtction is b. (ln(π) * [tex]\pi^(3x^2-7)[/tex]) * (6x) + (9 - x²) / (x²+9)² - 32 / sqrt(1 - (32x+1)²).

a. y = sin(7x)/tan(3x)

To differentiate this function, we can use the quotient rule, which states that if we have a function in the form f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative of f(x) is given by:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))²

In this case, g(x) = sin(7x) and h(x) = tan(3x). Let's differentiate both g(x) and h(x) first:

g'(x) = d/dx [sin(7x)] = 7 * cos(7x)

h'(x) = d/dx [tan(3x)] = 3 * sec²(3x)

Now we can substitute these derivatives into the quotient rule formula:

y' = [(7 * cos(7x) * tan(3x)) - (sin(7x) * 3 * sec²(3x))] / (tan(3x))²

Simplifying further, we get:

y' = [7 * cos(7x) * tan(3x) - 3 * sin(7x) * sec²3x)] / [tan²(3x)]

b. y = [tex]\pi^{(3x^2-7)[/tex] + x/(x²+9) + cos⁻¹(32x+1)

To differentiate this function, we can use the sum and chain rules. Let's differentiate each term separately:

For the first term, y₁ = [tex]\pi^{(3x^2-7)[/tex]:

y₁' = d/dx [[tex]\pi^{(3x^2-7)[/tex]]

Using the chain rule, the derivative is:

y₁' = (ln(π) * [tex]\pi^{(3x^2-7)[/tex]) * (6x)

For the second term, y₂ = x/(x²+9):

y₂' = d/dx [x/(x²+9)]

Using the quotient rule, the derivative is:

y₂' = [(1 * (x²+9)) - (x * 2x)] / (x²+9)²

Simplifying further, we get:

y₂' = (9 - x²) / (x²+9)²

For the third term, y₃ = cos⁻¹(32x+1):

y₃' = d/dx [cos⁻¹(32x+1)]

Using the chain rule, the derivative is:

y₃' = -32 / sqrt(1 - (32x+1)²)

Now, we can add all the derivatives together to find the derivative of the function:

y' = y₁' + y₂' + y₃'

y' = (ln(π) * [tex]\pi^{(3x^2-7)[/tex])) * (6x) + (9 - x²) / (x²+9)² - 32 / sqrt(1 - (32x+1)²)

The complete question is:
a. Differentiate the function: [tex]y=\frac{sin(7x)}{tan(3x)}[/tex].

b. Differentiate the function: [tex]\pi^{(3x^2-7)[/tex] + x/x²+9+cos⁻¹(32x+1)

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