Question

A contractor purchases a backhoe for $36600. Fuel and standard maintenance cost $5.99 per hour, and the operator is paid $12.91 per hour. A. Write a cost function for the cost C operating the backhoe for x hours. Be sure to include the purchase price in the cost function. Cost equation: C = ___________ dollars B. If customers pay $34.8 per hour for the contractors backhoe service, write the revenue function R for the amount of revenue gained from x hours of use. Revenue equation: R = _______ dollars C. Write the profit function P for the amount of profit gained from x hours of use. Profit equation: P = _______ dollars D. Use part C to find the number of hours the backhoe must be used to break even (when profit = 0). Please make sure the answer is correct to at least the nearest whole number. Break even = _________ hours

Answer 1

Given

Backhoe purchased for $36600

Maintenance cost of $5.99 per hour

Operator paid at $12.91 per hour

A. Writing a cost function.

$\begin{gathered} \text{Let }x\text{ be the number of hours for operating the backhoe} \\ \\ \text{Combine the hourly rate of maintenance cost and operator fee} \\ C(x)=(5.99+12.91)x+36600 \\ C(x)=(18.9)x+36600 \\ \\ \text{Therefore, the cost function is} \\ C(x)=18.9x+36600 \end{gathered}$

B. Writing the revenue function.

$\begin{gathered} \text{GIven that the customer pays for \ 34.8 per hour, then the revenue is calculated as} \\ R(x)=34.8x \end{gathered}$

C. Writing the Profit function

$\begin{gathered} \text{The profit shall be derived as Revenue minus Cost of operating} \\ P(x)=R(x)-C(x) \\ P(x)=34.8x-(18.9x+36600) \\ P(x)=34.8x-18.9x-36600 \\ \\ \text{Therefore, the profit function is} \\ P(x)=15.9x-36600 \end{gathered}$

D. Find the number of hours to breakeven

$\begin{gathered} \text{To breakeven, set }P(x)=0,\text{ and solve for }x \\ \\ P(x)=15.9x-36600 \\ 0=15.9x-36600 \\ \\ \text{Subtract }15.9x\text{ to both sides} \\ 0-15.9x=\cancel{15.9x-15.9x}-36600 \\ -15.9x=-36600 \\ \frac{-15.9x}{-15.9}=\frac{-36600}{-15.9} \\ \frac{\cancel{-15.9}x}{\cancel{-15.9}}=\frac{-36600}{-15.9} \\ x=2301.88679 \\ \\ \text{Rounding to the nearest whole number, the number of hours to breakven is} \\ x=2302\text{ hours} \end{gathered}$