Thread: Calculus Optimization Problem

I know i need two equations and i end up finding the derivative, but i am really struggling with this problem

The owners of a potato processing plant permit nearby farmers to rent their machines during the weekends. The rental fee is $50 per machine plus $30 per hour to rent the plant regardless of the number of machines used. How many machines should a farmer rent to minimize his cost for processing 25,000 bushels of potatoes if each machine can produce 75 bushels per hour?

Originally Posted by emaloof

I know i need two equations and i end up finding the derivative, but i am really struggling with this problem

The owners of a potato processing plant permit nearby farmers to rent their machines during the weekends. The rental fee is $50 per machine plus $30 per hour to rent the plant regardless of the number of machines used. How many machines should a farmer rent to minimize his cost for processing 25,000 bushels of potatoes if each machine can produce 75 bushels per hour?

let $\displaystyle n$ = number of machines rented

number of hours needed for $\displaystyle n$ machines to process 25000 bushels = $\displaystyle \dfrac{25000}{75n}$

cost as a function of the number of machines rented ...

$\displaystyle C(n) = 50n + 30 \cdot \dfrac{25000}{75n}$

skeeter, thank you so much! I am still having trouble finding the derivative of that equation you gave me. I am supposed to find the derivative correct?

Originally Posted by emaloof

skeeter, thank you so much! I am still having trouble finding the derivative of that equation you gave me. I am supposed to find the derivative correct?

that would be correct ... try simplifying the numbers in the cost function first with some arithmetic, then find the derivative and set it equal to 0 to find the critical value of n