Identify the equation of the line that passes through the pair of points (−3, 6) and (−5, 9) in slope-intercept form.

a) When the radius is 4 cm, it is changing at a rate of 1/(4π) cm/s.

b) When the circumference is 3 cm, the radius is changing at a rate of 2/3 cm/s.

a. Given that the area of a circle increases at a rate of 2 cm²/s, let's denote this rate as dA/dt.

The formula for the area of a circle is A = πr²,

where A is the area and r is the radius.

We want to find the rate at which the radius is changing, or dr/dt, when the radius is 4 cm.

Using implicit differentiation with respect to time t, we get:

dA/dt = d(πr²)/dt 2 = 2πr(dr/dt)

Now, we'll plug in the radius value of 4 cm:

2 = 2π(4)(dr/dt)

Solving for dr/dt, we get:

dr/dt = 1/(4π) cm/s

b. We are given the circumference, which is 3 cm.

The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.

First, we need to find the radius when the circumference is 3 cm: 3 = 2πr r = 3/(2π)

Now, we'll plug this value for the radius back into the formula from part a:

2 = 2π(3/(2π))(dr/dt)

Solving for dr/dt, we get:

dr/dt = 2/3 cm/s

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The complete equation of two squares to find the product of 22 and 18 is (22 + 18)(22 - 18) = 396

When we can interpret an expression as the difference of two perfect squares, i.e. a2-b2, we can factor it as (a+b)(a-b).

To use the difference of two squares to find the product of 22 and 18:

First, find the average of the two numbers:

(22 + 18) ÷ 2 = 20

Then, find the difference between the two numbers:

22 - 18 = 4

Now we can write:

(20 + 4)(20 - 4) = 24 × 16 = 384

But we need to add the extra 12 to get 396:

(20 + 4)(20 - 4) + 12 = 396

So the completed equation is:

(22 + 18)(22 - 18) = 396

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The debt increased between 1996 and 2003. Then the national debt increased by approximately $4,903.73 billion between 1996 and 2003.

To find how much the debt increased between 1996 and 2003, we need to find the value of the function D'(t) for t=7 (since 2003 is 7 years after 1996).

D'(t)=858.29+819.48t-184.32t^2+12.12t^3

D'(7)=858.29+819.48(7)-184.32(7^2)+12.12(7^3)
D'(7)=858.29+5,736.36-8,132.32+3,458.68
D'(7)=1,921.01

Therefore, the annual rate of change in the national debt in 2003 was $1,921.01 billion per year.

To find how much the debt increased between 1996 and 2003, we need to integrate the function D'(t) from t=1 to t=7:

∫(D'(t))dt = ∫(858.29+819.48t-184.32t^2+12.12t^3)dt
= 858.29t + 409.74t^2 - 61.44t^3 + 3.03t^4 + C

where C is the constant of integration.

Evaluating this expression at t=7 and t=1 and taking the difference, we get:

(858.29(7) + 409.74(7)^2 - 61.44(7)^3 + 3.03(7)^4 + C) - (858.29(1) + 409.74(1)^2 - 61.44(1)^3 + 3.03(1)^4 + C)
= 6,111.09 - 1,207.36 = 4,903.73

Therefore, the national debt increased by approximately $4,903.73 billion between 1996 and 2003.

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Keegan will have approximately $878.85 in the account after 5 years.

What is Compound interest ?

Compound interest is the interest that is earned not only on the initial amount of money invested (known as the principal), but also on any interest earned on that principal over time. In other words, compound interest is interest on interest.

The compound interest formula is given by:

A = P[tex](1 + r/n)^{nt}[/tex]

where A is the amount after t years, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

In this case, Keegan deposited $675, the annual interest rate is 4.8%, the interest is compounded quarterly, and the investment is for 5 years. Therefore, we can plug in these values into the formula to get:

A = 675[tex](1 + 0.048/4)^{20}[/tex]

A = 675[tex](1.012)^{20}[/tex]

A ≈ $878.85

Therefore, Keegan will have approximately $878.85 in the account after 5 years.

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Given the sequence an = n^7/n^4, The sequence is = divergent. So, the answer is that the sequence is divergent.

The sequence an = n^7/n^4 can be simplified by cancelling out the common factor of n^4 in the numerator and denominator. This yields an = n^3.

Therefore, the sequence is simply the cubes of the positive integers, or 1, 8, 27, 64, 125, 216, ... and so on. This sequence grows without bound, as n gets larger, so it does not converge to any finite limit. In other words, the sequence an does not have a limit, and it is divergent.

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In a school, a teacher surveys 50 randomly selected 6th grade students and 50 randomly selected 7th grade students. c) More seventh-graders than sixth-graders enjoy horror films.

The first statement is false as the total 6th graders which like horror and comedy movie is 19 + 9 = 28 students which is more than 6th graders who like action movies which is 22, hence the first statement is false. this is interpreted from given data set.

The second statement is also false as it says that 6th graders prefer comedy films to action films, whereas 7th graders prefer action films but from the data given, it can be seen that the number of 6th graders who like comedy films is same as the number of 7th graders who like action movies which is 19, hence statement is false.

The third statement is true as 6th graders who like horror movies is 9 while 7th graders who like horror movies is 14 and hence, the statement is true.

Fourth statement is also false as 17, 7th graders like comedy movies in contrast to 14, 7th graders who like horror movies.

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Correct question:

A teacher conducts a survey of 50 randomly selected 6th grade and 50 randomly selected 7th grade students in a school. The survey asks the students about the type of movies they like to watch. The table shows the number of students who selected each type of movie. Select the correct statement.

a) 6th graders like action movies more than horror and comedy movies.

b) 6th graders like comedy movies more than 7th graders like action movies.

c) More 7th graders than 6th graders like horror movies.

d) 7th graders like horror movies more than comedy movies.

Answer:

B

Step-by-step explanation:

Lisa's first step is to distribute, so you have to distribute all numbers.

(3+6x)-2(x+1)+5

3(1+2x)-2)x+1)+5

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The tax on a bicycle costing $700, with the tax remaining the same as a bicycle costing $400 with a tax of $32, will be $56.

To calculate the tax on a bicycle costing $700, we need to know the percentage of tax charged on the $400 bicycle. The tax on the $400 bicycle is $32. To find the tax rate, we divide the tax by the cost of the bicycle and multiply by 100 to get a percentage.

tax rate = (tax / cost of bicycle) x 100%

tax rate = (32 / 400) x 100%

tax rate = 8%

Therefore, the tax rate is 8%. We can use this tax rate to calculate the tax on a bicycle costing $700.

tax on $700 bicycle = (tax rate / 100) x cost of bicycle

tax on $700 bicycle = (8 / 100) x $700

tax on $700 bicycle = $56

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We can replace the ellipsis with -49a4 to get: (7/4)a 10 - 2 = -49a4 - b 4, missing monomial is -49a4.

To find the missing monomial, we need to use the distributive property to expand the left side of the equation:

(... - b 4)(64 + ....) = 121a 10 – 68

Expanding the left side gives:

64(... - b 4) + ....(... - b 4) = 121a 10 – 68

Now we need to find a monomial to replace the ellipsis that will make the left side of the equation equal to the right side of the equation, no matter what value is substituted for a and b.

Looking at the constants in the equation, we can see that 121 and 68 have a common factor of 17. Therefore, we can divide both sides of the equation by 17 to simplify the coefficients:

7a 10 - 4 = 4(... - b 4) + 4

Simplifying further, we get:

7a 10 - 8 = 4(... - b 4)

Dividing both sides by 4, we get:

(7/4)a 10 - 2 = ... - b 4

Now we need to find a monomial to replace the ellipsis that will make the left side of the equation equal to the right side of the equation, no matter what value is substituted for a and b.

Since we want the left side of the equation to have a degree of 4 (because the right side has a term of -b4), we need to choose a monomial of degree 4 that will cancel out the terms of degree 10 on the left side. One possible monomial is:

-49a4

Therefore, we can replace the ellipsis with -49a4 to get: (7/4)a 10 - 2 = -49a4 - b 4

So the missing monomial is -49a4.

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The intervals where f is decreasing are given as follows:

None of the above.

Hence the decreasing intervals of the function are given as follows:

Which are none of the options given in the problem.

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The area of the shaded region is 422.8 m², rounded to the nearest tenth.

To find the area of the shaded region, we first need to determine the areas of the two shapes that make up the region. The first shape is a rectangle with dimensions of 18.6 m by 30 m, which has an area of:

Area of rectangle = length x width = 18.6 m x 30 m = 558 m²

The second shape is a semi-circle with a diameter of 18.6 m, which has a radius of 9.3 m. The area of a semi-circle is half the area of a full circle, so we can use the formula for the area of a circle to find the area of the semi-circle:

Area of semi-circle = (1/2) x π x r² = (1/2) x π x 9.3² = 135.2 m²

To find the area of the shaded region, we need to subtract the area of the semi-circle from the area of the rectangle:

Area of shaded region = Area of rectangle - Area of semi-circle
Area of shaded region = 558 m² - 135.2 m²
Area of shaded region = 422.8 m²

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For the following expression arcsin (sin (-57π/10)) is 3π/10.

The function arcsin(x) gives the angle in radians whose sine is x.

In this problem, we need to find the angle whose sine is equal to the sine of -57π/10.

First, we need to simplify -57π/10 to an angle in the range [-π/2, π/2] since the sine function has a range of [-1, 1]. To do this, we use the fact that sine has a period of 2π,

which means that sin(-57π/10) = sin((-57π/10) + 4π) = sin(3π/10).

So we need to find the angle θ such that sin(θ) = sin(3π/10).

Since sine is an odd function, we know that sin(-θ) = -sin(θ), so we can also say that sin(θ) = sin(-3π/10).

Therefore, there are two possible angles that satisfy the equation: θ = 3π/10 or θ = -3π/10.

However, since the range of the arcsine function is [-π/2, π/2], only the angle in that range that satisfies the equation is θ = 3π/10.

Therefore, we can write:

arcsin(sin(-57π/10)) = arcsin(sin(3π/10)) = 3π/10

The answer is 3π/10.

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The line crosses the y-axis at -9 and has a slope of 4, meaning it increases 4 units in the y-direction for every 1 unit increase in the x-direction.

A graph is a visual representation of data, typically involving the use of points, lines, and curves to show how different values are related to each other. Graphs can be used to display a wide range of information, from mathematical functions and scientific data to business trends and social networks.

Here,

To graph y = 4x - 9, we can start by plotting the y-intercept, which is -9 on the y-axis. Then, we can use the slope of 4 to find other points on the line.

Slope = 4 can be written as rise/run = 4/1. This means that for every increase of 1 in the x-direction, the y-value increases by 4.

Starting from the y-intercept (-9,0), we can go up 4 units and to the right 1 unit to get the point (1,-5). We can continue this pattern and plot more points on the line, or we can simply draw a straight line through the y-intercept and (1,-5) to represent the equation y = 4x - 9.

Here's what the graph looks like:

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Answer:

C) [tex]\sf \sqrt{45}[/tex]

Step-by-step explanation:

Pythagorean theorem:

               AB = 6 units

              BC = 3 units

AC is hypotenuse and AB is the base and BC is the altitude.

Hypotenuse² = base² + altitude²

               AC² = AB² + BC²

                      [tex]\sf = 6^2 + 3^2\\\\ = 36 + 9\\\\ = 45[/tex]

               [tex]\sf AC= \sqrt{45}[/tex]

A good description of the cross section shown that is parallel to the edge of the pyramid that measures 5​ millimeters is a triangle with base of 5 millimeters and height of 16 millimeters.

In Mathematics and Geometry, a square pyramid can be defined as a type of pyramid that has a square base, four (4) triangular sides, five (5) vertices, and eight (8) edges.

In Mathematics and Geometry, a triangle can be defined as a two-dimensional (2D) geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.

In this context, we can reasonably infer and logically deduce that the edge of the prism that measures 5 millimeters represents a triangle with base of 5 millimeters and height of 16 millimeters.

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Ben could have saved $0.15 per gallon if he had gone to the gas station across the street.

If Ben filled his 20-gallon tank of gas with regular fuel for $59.80, then the cost per gallon can be found by dividing the total cost by the number of gallons:

cost per gallon = total cost / number of gallons

cost per gallon = $59.80 / 20 gallons

cost per gallon = $2.99/gallon

If the gas station across the street sold regular fuel for $2.84 a gallon, then the amount Ben could have saved per gallon is:

savings per gallon = cost per gallon at initial station - cost per gallon at other station

savings per gallon = $2.99/gallon - $2.84/gallon

savings per gallon = $0.15/gallon

Therefore, Ben could have saved $0.15 per gallon if he had gone to the gas station across the street.

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To prove the trigonometric identity csc(x) - sin(x) - cos(x)cos(x), we will start by simplifying the left-hand side of the equation using trigonometric definitions and identities.

Recall that csc(x) = 1/sin(x). We will use this definition to rewrite the left-hand side of the equation:

1/sin(x) - sin(x) - cos(x)cos(x)

Now, we will find a common denominator for the terms in the equation. In this case, the common denominator is sin(x). To do this, we will multiply sin(x) to the second term:

(1 - sin^2(x) - cos(x)cos(x)sin(x)) / sin(x)

Next, we will use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) in the expression:

(1 - (1 - cos^2(x)) - cos(x)cos(x)sin(x)) / sin(x)

Simplifying the expression, we get:

(cos^2(x) - cos(x)cos(x)sin(x)) / sin(x)

Now, we can factor out cos(x) from the numerator:

cos(x)(cos(x) - sin(x)) / sin(x)

This expression is equivalent to the given identity, so we have proven the trigonometric identity:

csc(x) - sin(x) - cos(x)cos(x) = cos(x)(cos(x) - sin(x)) / sin(x)

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The magnitude of vector v is:
|v| = 7

The magnitude of v is simply the length of the vector v, which can be found using the Pythagorean theorem. The vector v is given as v = 7i.

To find the magnitude of v (|v|), use the formula:
|v| = √(x² + y²)

where x and y are the components of the vector v. In this case, x = 7 (from 7i) and y = 0 (since there is no j component).

Now, plug in the values of x and y into the formula:
|v| = √(7² + 0²)
|v| = √(49 + 0)
|v| = √(49)
|v| = 7

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Ms. Brock can now fill up to 88 spray bottles with her preparation.

It is calculated that:

hence we have that

1 gallon = 128 fluid ounces

1 pint = 16 fluid ounces  

cross multiplying

16 fluid ounces x 1 gallon = 128 fluid ounces x 1 pint

1 gallon = (128 fluid ounces x 1 pint) / 16 fluid ounces

1 gallon = 8 pints

Consequently, there are 8 pints in one full gallon.

Ms. Brock blended 1 gallon of cleaner with 10 gallons of water, thus resulting in 11 gallons.

To figure out the total number of pints Ms. Brock has in her solution, we must multiply 11 gallons by 8 pints/gallon:

11 gallons x 8 pints/gallon = 88 pints

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Answer:

$64

Step-by-step explanation:

This would be the answer because if we figure out what needs to be added to -21 to reach 43 we get 64.

(-21+64 - $43.)

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Sin theta = -5/sqrt(26) Is the correct option , To find the six trigonometric functions of theta, we need to first find the values of x and y on the terminal side of theta. From the given equation, 5x + y = 0 and x ≥ 0, we can solve for y in terms of x:

y = -5x

Since x ≥ 0, we know that (x, y) lies in the fourth quadrant. We can now use the Pythagorean theorem to find the length of the hypotenuse:

r =[tex]sqrt(x^2 + y^2) = sqrt(x^2 + (-5x)^2) = sqrt(26x^2)[/tex]

Now we can find the six trigonometric functions:

sin(theta) = y/r = [tex]-5x/sqrt(26x^2) = -5/sqrt(26)[/tex]

cos(theta) = x/r =[tex]x/sqrt(26x^2) = 1/sqrt(26)[/tex]

tan(theta) = y/x = -5x/x = -5

csc(theta) = r/y = [tex]sqrt(26x^2)/(-5x) = -sqrt(26)/5[/tex]

sec(theta) = r/x = sqrt(26x^2)/x = sqrt(26)/1 = sqrt(26)

cot(theta) = 1/tan(theta) = -1/5

Therefore, the exact values of the six trigonometric functions of theta are:

sin(theta) = -5/sqrt(26)

cos(theta) = 1/sqrt(26)

tan(theta) = -5

csc(theta) = -sqrt(26)/5

sec(theta) = sqrt(26)

cot(theta) = -1/5

Answer: A. sin theta = -5/sqrt(26)

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(a) The triangle is a right triangle when x is equal to 24.

(b) The side lengths do not form a Pythagorean triple.

(a) To determine when the triangle is a right triangle, we can apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Using the given side lengths, we have:

5^2 + 7^2 = x^2

Simplifying the equation:

25 + 49 = x^2

74 = x^2

To find the value of x, we take the square root of both sides:

x = √74

Approximating the square root of 74, we get:

x ≈ 8.60

Therefore, when x is approximately 8.60, the triangle is a right triangle.

(b) For the side lengths to form a Pythagorean triple, they must satisfy the condition of the Pythagorean theorem. In this case, we have:

5^2 + 7^2 = x^2

Simplifying the equation:

25 + 49 = x^2

74 = x^2

Since the sum of the squares of the two smaller sides (25 + 49 = 74) is not equal to the square of the longest side (x^2 = 74), the side lengths of 5, 7, and x do not form a Pythagorean triple.

In conclusion, the triangle is a right triangle when x is equal to approximately 8.60, and the side lengths do not form a Pythagorean triple.

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Projection of vector u onto vector v is approximately (1.33, 0.76, -0.57).

We'll use the following formula:

projection of u onto v = (u•v / ||v||²) * v

First, we need to calculate the dot product (u•v) and the magnitude squared (||v||²) of vector v.

1. Dot product (u•v):
u•v = (5 * -7) + (-9 * -4) + (-5 * 3) = -35 + 36 - 15 = -14

2. Magnitude squared (||v||²):
||v||^2 = (-7)² + (-4)² + (3)² = 49 + 16 + 9 = 74

Now, we'll plug these values into the projection formula:

projection of u onto v = (-14 / 74) * v

We'll multiply each component of vector v by the scalar (-14/74):

projection of u onto v = (-14/74) * (-7, -4, 3) = (1.33, 0.76, -0.57)

So, the projection of vector u onto vector v is approximately (1.33, 0.76, -0.57).

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Ted Pappas' tax rate to the nearest tenth of a mill is approximately 87.6 mills.

First, let's determine the assessed value of the property. To do this, we'll multiply the market value by the rate of assessment:

Assessed Value = Market Value × Rate of Assessment
Assessed Value = $119,340 × 0.42
Assessed Value = $50,122.80

Now, we need to find the tax rate in mills. One mill is equal to $1 per $1,000 of assessed value. To find the tax rate, we'll divide the yearly real estate taxes by the assessed value and multiply by 1,000:

Tax Rate (in mills) = (Yearly Real Estate Taxes / Assessed Value) × 1,000
Tax Rate = ($4,388.65 / $50,122.80) × 1,000
Tax Rate ≈ 87.6 mills

Therefore, Ted Pappas' tax rate to the nearest tenth of a mill is approximately 87.6 mills.

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After the duration of 2 hours the distance between the first pair and the second pair is 3.07 km, under the condition that the second pair decide to head 60° west of north travelling at the same pace.

In order to evaluate the distance between two points with given coordinates, we can apply the distance formula. The distance formula is

d = √ [ (x₂ − x₁)² + (y₂ − y₁)² ]

Here,

(x₁, y₁) and (x₂, y₂) = coordinates of the two points.

For the given case, we can consider that the first pair of team members start at the origin (0, 0) and cover the distance towards  north for 2 hours at a pace of 4 km/h.

Hence, their final position is (0, 8).

The second pair of team members take  the origin (0, 0) and travel 60° west of north for 2 hours at a pace of 4 km/h.

Now to evaluate their final position, we have  to find their coordinates. Let us consider their final position (x, y).

We can apply trigonometry to find x and y.

The angle between their direction of travel and the y-axis is 60°.

sin(60°) = y / d

cos(60°) = x / d

Here,

d = distance travelled by the second pair of team members.

It is given that they travelled for 2 hours at a pace of 4 km/h.

d = 2 hours × 4 km/h

= 8 km

Staging this value into the above equations

y = d × sin(60°) = 8 km × sin(60°)

≈ 6.93 km

x = d × cos(60°) = 8 km × cos(60°)

≈ 4 km

Hence, the final position regarding the second pair of team members is approximately (4 km, 6.93 km).

Now we can apply the distance formula to evaluate the distance between the two pairs of team members

d = √ [ (x₂ − x₁)² + (y₂ − y₁)² ]

d = √ [ (4 − 0)² + (6.93 − 8)² ]

d ≈ 3.07 km

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The given radian measure -1.7 pi is equivalent to -306 degrees.

The correct answer is option (b), Negative 306 degrees. This conversion takes into account the negative sign of the radian measure, resulting in a negative degree measure to convert a radian measure to a degree measure, we use the conversion factor that 180 degrees is equal to π radians.

Given the radian measure -1.7π, we can calculate the corresponding degree measure by multiplying -1.7π by the conversion factor:

Degree measure = (-1.7π) * (180 degrees / π)

The π in the numerator and denominator cancels out, resulting in:

Degree measure = -1.7 * 180 degrees

Calculating the value, we have:

Degree measure = -306 degrees

Therefore, the correct answer is option b) Negative 306 degrees.

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The admission price that will yield maximum revenue is $6.

To determine the admission price that will yield maximum revenue, we'll first find the linear demand function using the given data points: ($7, 1000) and ($6, 1200).

Let x represent the admission price and y represent the number of customers per day. We can calculate the slope (m) using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the given data points:

m = (1200 - 1000) / (6 - 7) = 200 / (-1) = -200

Now, we have the slope and a point, so we can use the point-slope form to find the linear demand function:

y - y1 = m(x - x1)

Using the point ($7, 1000):

y - 1000 = -200(x - 7)

Now, let's rewrite the equation to the slope-intercept form (y = mx + b):

y = -200x + 2400

The revenue (R) is equal to the product of the admission price (x) and the number of customers (y):

R = xy

Substitute the linear demand function (y = -200x + 2400) into the revenue equation:

R = x(-200x + 2400)

To maximize the revenue, we need to find the vertex of the parabola represented by this equation. The x-coordinate of the vertex is given by:

x_vertex = -b / 2a

In this case, a = -200 and b = 2400:

x_vertex = -2400 / (2 * -200) = 6

The admission price that will yield maximum revenue is $6.

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(a) The critical numbers are x = 0 and x = 6.

(b) The function is increasing on the interval (0, 6) and decreasing on the intervals (-∞, 0) and (6, ∞).

(a) To find the critical numbers of f(x), we need to find the values of x where f'(x) = 0 or f'(x) does not exist.

f'(x) = 12x² - 72x

Setting f'(x) = 0, we get:

12x² - 72x = 0

12x(x - 6) = 0

So, the critical numbers are x = 0 and x = 6.

(b) To determine where the function is increasing or decreasing, we need to examine the sign of f'(x) on different intervals.

For x < 0, f'(x) = 12x² - 72x < 0, which means the function is decreasing on (-∞, 0).

For 0 < x < 6, f'(x) = 12x² - 72x > 0, which means the function is increasing on (0, 6).

For x > 6, f'(x) = 12x² - 72x < 0, which means the function is decreasing on (6, ∞).

So, the function is increasing on the interval (0, 6) and decreasing on the intervals (-∞, 0) and (6, ∞).

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The interest he had received in total after eight years is 1396.59 kroner

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

Principal, P = 7000

Rate, r = 2.3%

Time, t = 8

The interest he had received in total after eight years is calculated as

I = P(1 + r)^t - P

Substitute the known values in the above equation, so, we have the following representation

I = 7000(1 + 2.3%)^8 - 7000

Evaluate

I = 1396.59

Hence, the total interest is 1396.59 kroner

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