# Thread: First year optimizing functions

1. ## First year optimizing functions

I am attempting to do this question, but every time I look at it, I just get more confused. I have no idea how to even start. I'm especially confused about how to show the relationship between the values. Please help =/

The cost of fuel (dollars/hour) to propel a ferry boat through water is proportional to the cube of the speed of the ferry boat. A certain ferry boat uses $100 of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the operational cost of running the ferry is$675 per hour. At what speed should the ferry boat cruise at to minimize the cost per mile traveled?

2. ## Re: First year optimizing functions

Originally Posted by jenniferz3
I am attempting to do this question, but every time I look at it, I just get more confused. I have no idea how to even start. I'm especially confused about how to show the relationship between the values. Please help =/

The cost of fuel (dollars/hour) to propel a ferry boat through water is proportional to the cube of the speed of the ferry boat. A certain ferry boat uses $100 of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the operational cost of running the ferry is$675 per hour. At what speed should the ferry boat cruise at to minimize the cost per mile traveled?
$\displaystyle F = kv^3$ , where $\displaystyle F$ is fuel cost in dollars/hr

$\displaystyle 100 = k \cdot 10^3$

$\displaystyle k = \frac{1}{10}$

total cost in dollars/hr = operational cost in dollars/hr + fuel cost in dollars/hr

$\displaystyle C = 675 + \frac{v^3}{10}$ , where $\displaystyle C$ is total cost in dollars/hr

finally, note that cost/mile = (cost/hr)/(miles/hr) ... optimize.

3. ## Re: First year optimizing functions

Thank you so much! Worked out perfectly. D