Question 1

Find the vertex of $f ( x ) = 5 x ^ { 2 } - 7 x + 2$ .

Accepted Answer

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Question 2

Suppose that a car rental agency charges a fixed amount per day plus an amount per mile for renting a car. Heidi rented a car one day and paid $110.00 for 300 miles. On another day she rented a car from the same agency and paid $185.00 for 550 miles. Determine the linear function that the agency could use to determine its daily rental charges. Please enter your answer in the form f ( x ) = ax + b .

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Question 3

The height of a projectile fired vertically into the air (neglecting air resistance) at an initial velocity of 144 feet per second is a function of the time and is given by the equation f ( x ) = 144 x $-$ 16 x 2 , where x represents the time. Find the highest point reached by the projectile.

Accepted Answer

To find the highest point reached by the projectile, we need to find the vertex of the parabola represented by the function f(x). The x-coordinate of the vertex is given by:
x = -b / 2a
where a = -16 and b = 144. Substituting the values, we get:
x = -144 / 2(-16) = 4.5
So, the highest point is reached after 4.5 seconds. Substituting this value back into the function, we get:
f(4.5) = 144(4.5) - 16(4.5)^2 = 324
Therefore, the highest point reached by the projectile is 324 feet.

Question 4

A motel advertises that it will provide dinner, a dance, and drinks at $30 per couple for a New Year's Eve party. It must have a guarantee of 60 couples. Furthermore, it will agree that for each couple in excess of 60, it will reduce the price per couple for all attending by $0.50. How many couples will it take to maximize the motel's revenue?

Accepted Answer

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Question 5

The linear depreciation method assumes that an item depreciates the same amount each year. Suppose a new piece of machinery costs $22,500 and it depreciates $1,450 each year for t years. Find the value of the machinery after 8 years.

Accepted Answer

Since the machinery depreciates $1,450 each year for t years, we can create the linear equation y = 22,500 - 1,450t, where y represents the value of the machinery after t years.

To find the value after 8 years, we substitute t = 8 into the equation:

y = 22,500 - 1,450(8) = 10,900

Therefore, the value of the machinery after 8 years is $10,900. Hence, the best choice is C.

Question 6

Suppose that a car rental agency charges a fixed amount per day plus an amount per mile for renting a car. Heidi rented a car one day and paid $60.00 for 200 miles. On another day she rented a car from the same agency and paid $90.00 for 350 miles. Determine the linear function that the agency could use to determine its daily rental charges.

Accepted Answer

Let x be the number of miles driven and y be the total cost. We can set up two equations with the given information:

60 = a + 200b     (where a is the fixed daily rental charge and b is the cost per mile)
90 = a + 350b

Solving for a and b using either substitution or elimination, we get:
a = 20 and b = 0.2

Therefore, the linear function for determining the daily rental charge is:
f(x) = 0.2x + 20.

Question 7

Find the x intercepts of $f ( x ) = 3 x ^ { 2 } + 20 x - 7$ .

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Question 8

Three hundred meters of fencing is available to enclose a rectangular playground. What should be the dimensions of the playground to maximize the area?

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Question 9

Suppose that the equation p ( x ) = $-$ 2 x 2 + 280 x $-$ 2,540, where x represents the number of items sold, describes the profit function for a certain business. How many items should be sold to maximize the profit? __________ items

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Question 10

Find two numbers whose sum is 48 and whose product is a maximum.

Accepted Answer

The product of two numbers with a fixed sum is maximized when the numbers are equal, so 24 and 24 give the maximum product.

Question 11

Suppose that the equation p ( x ) = $-$ 5 x 2 + 600 x $-$ 4,705, where x represents the number of items sold, describes the profit function for a certain business. How many items should be sold to maximize the profit?

Accepted Answer

To maximize profit, we need to find the vertex of the parabola represented by p(x). The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = -5 and b = 600.

x = -600/(2*(-5)) = 60

Therefore, the business should sell 60 items to maximize profit.

Question 12

Find two numbers whose sum is 60 and whose product is a maximum. Please enter your answer as two numbers separated by a comma, without the units.

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The answer of Find two numbers whose sum is 60...

Question 13

The linear depreciation method assumes that an item depreciates the same amount each year. Suppose a new piece of machinery costs $25,500 and it depreciates $1,750 each year for t years. Set up a linear function that yields the value of the machinery after t years. Please enter your answer in the form f ( t ) = at + b .

Accepted Answer

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Question 14

Zack wants to sell five items that cost him $1.50, $2.70, $5.20, $13.00, and $17.70. He wants to make a profit of 70% of the cost. Create a function that you can use to determine the selling price of each item, and then use the function to calculate each selling price.

Accepted Answer

To calculate the selling price, we need to add 70% of the cost to the cost. This can be done using the following formula:
selling price = cost + 0.7*cost = 1.7*cost

Using this formula, we can calculate the selling price for each item:
- Item 1: 1.7*1.5 = $2.55
- Item 2: 1.7*2.7 = $4.59
- Item 3: 1.7*5.2 = $8.84
- Item 4: 1.7*13 = $22.10
- Item 5: 1.7*17.7 = $30.09

Therefore, option B is the best choice as it provides the correct selling prices for each item.

Question 15

The height of a projectile fired vertically into the air (neglecting air resistance) at an initial velocity of 64 feet per second is a function of the time and is given by the equation f ( x ) = 64 x $-$ 16 x 2 , where x represents the time. Find the highest point reached by the projectile. __________ feet

Accepted Answer

The answer of The height of a projectile fired vertically...

Question 16

A motel advertises that it will provide dinner, a dance, and drinks at $40 per couple for a New Year's Eve party. It must have a guarantee of 60 couples. Furthermore, it will agree that for each couple in excess of 60, it will reduce the price per couple for all attending by $0.10. How many couples will it take to maximize the motel's revenue? __________ couples

Accepted Answer

The answer of A motel advertises that it will provide...

Question 17

The cost for burning a 75-watt bulb is given by the function c ( h ) = 0.0045 h , where h represents the number of hours that the bulb burns. How much does it cost to burn a 75-watt bulb for 3 hours per night for a 31-day month? Express your answer to the nearest cent.

Accepted Answer

The bulb burns for 3 hours per night, so it burns for a total of 3 x 31 = 93 hours in the month. 
Using the given function, we can calculate the total cost: 
c(h) = 0.0045h 
c(93) = 0.0045(93) = 0.4185 
Rounding to the nearest cent, the cost is $0.42. 
Therefore, the best choice is B.

Question 18

The linear depreciation method assumes that an item depreciates the same amount each year. Suppose a new piece of machinery costs $46,000 and it depreciates $850 each year for t years. Find the value of the machinery after 4 years. $ __________

Accepted Answer

The answer of The linear depreciation method assumes that an...

Question 19

The cost for burning a 75-watt bulb is given by the function c ( h ) = 0.0045 h , where h represents the number of hours that the bulb burns. How much does it cost to burn a 75-watt bulb for 4 hours per night for a 30-day month? Express your answer to the nearest cent. $ __________

Accepted Answer

The answer of The cost for burning a 75-watt bulb...

Question 20

The linear depreciation method assumes that an item depreciates the same amount each year. Suppose a new piece of machinery costs $32,000 and it depreciates $1,650 each year for t years. Set up a linear function that yields the value of the machinery after t years.

Accepted Answer

The linear function that yields the value of the machinery after t years is given by:
f ( t ) = 32,000 - 1,650 t
This is because the machinery depreciates by $1,650 each year, so after t years, the total depreciation will be $1,650 times t. We subtract this from the original cost of $32,000 to get the value of the machinery after t years. Hence, the correct answer is choice A.

Quiz 13: Functions

Find the vertex of f(x)=5x2−7x+2f ( x ) = 5 x ^ { 2 } - 7 x + 2f(x)=5x2−7x+2 .

The height of a projectile fired vertically into the air (neglecting air resistance) at an initial velocity of 144 feet per second is a function of the time and is given by the equation f ( x ) = 144 x −-− 16 x 2 , where x represents the time. Find the highest point reached by the projectile.