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Bea T. Howen, a sophomore college student, lost her scholarship after receiving a D in her "Music Appreciation" course. She decided to buy a snow plow to supplement her income during the winter months. It cost her $6312.50. Fuel and standard maintenance will cost her an additional $8.75 for each hour of use. Find the cost function C(x) associated with operating the snow plow for x hours. C(x)= If she charges $34.00 per hour write the revenue function R(x) for the amount of revenue gained from x hours of use. R(x)= Find the profit function P(x) for the amount of profit gained from x hours of use. P(x)= How many hours will she need to work to break even? hours

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MrRoyal

Answer:

[tex](a)\ C(x) = 6312.50 + 8.75x[/tex]

[tex](b)\ R(x) = 34x[/tex]

[tex](c)\ P(x) = 6312.50 -25.25x[/tex]

[tex](d)\ Break\ Even = 250\ hours[/tex]

Step-by-step explanation:

Solving (a): The cost function.

Given

[tex]Cost = \$6312.50[/tex] --- cost of saxophone

[tex]Additional = \$8.75[/tex] per hour

The cost function is: sum of the cost of the saxophone and the extra cost per hour.

If an hour costs 8.75, then x hours will cost 8.75x

So, the cost function is:

[tex]C(x) = Cost + Additional[/tex]

[tex]C(x) = 6312.50 + 8.75x[/tex]

Solving (b): The revenue function

Given

[tex]Charges = \$34.00[/tex] per hour

The revenue function is the product of the unit charge by the number of hours.

If in an hour, she charges $34.00, then in x hours, she will cost 34x

So, the revenue function is:

[tex]R(x) = Charges * Hours[/tex]

[tex]R(x) = 34.00 * x[/tex]

[tex]R(x) = 34x[/tex]

Solving (c): The profit function

This is the difference between the cost function and the revenue function

i.e.

[tex]P(x) = C(x) - R(x)[/tex]

So, we have:

[tex]P(x) = 6312.50 + 8.75x - 34x[/tex]

[tex]P(x) = 6312.50 -25.25x[/tex]

Solving (d): The break even hours.

To do this, we simply equate the cost function and the revenue function, then solve for x

i.e.

[tex]C(x) =R(x)[/tex]

[tex]6312.50 + 8.75x = 34x[/tex]

Collect like terms

[tex]6312.50 =- 8.75x + 34x[/tex]

[tex]6312.50 = 25.25x[/tex]

Solve for x

[tex]x = \frac{6312.50}{25.25}[/tex]

[tex]x = 250[/tex]

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