# Thread: Optimization problem

1. ## Optimization problem

A small business uses a minivan to make deliveries. The cost per hour for fuel is F=(v^2)/(360), where v is the speed of the minivan (in miles per hour). The driver is paid $10 per hour. Find the speed that minimizes the cost of a 110-mile trip. (Assume there are no costs other than fuel and wages.) I don't know how to set this particular problem up 2. Originally Posted by dassix A small business uses a minivan to make deliveries. The cost per hour for fuel is F=(v^2)/(360), where v is the speed of the minivan (in miles per hour). The driver is paid$10 per hour. Find the speed that minimizes the cost of a 110-mile trip. (Assume there are no costs other than fuel and wages.)

I don't know how to set this particular problem up
1. Costs for the fuel: $\displaystyle F(t)=\frac1{360} v^2 \cdot t$

Wages of the driver $\displaystyle w(t)=10 t$. Therefore: The total costs are:

$\displaystyle c= \frac1{360} v^2 \cdot t + 10t$

2. From the definition of speed you'll get:

$\displaystyle speed=\frac{distance}{time} \text{ that means: } v=\frac{110}t~\implies~t=\frac{110}v$

3. Plug in this term into the equation of the total costs:

$\displaystyle c(v)=\frac{110}{360} v+\frac{1100}v$

4. Differentiate c(v) wrt v and solve for v the equation c'(v) = 0.