1. ## More Optimization

Problem: The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed v. A certain ferry boat uses $196 worth of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the cost of running this ferry (labor, maintenance, and so on) is$1323 per hour.

(a) Find the proportionality constant for the cost of the fuel per hour.

(b) Express the cost per mile in terms of v only.

(c) At what speed should the boat travel so as to minimize the cost per mile traveled?

I really don't know how to set up an equation for this, or how to find the proportionality constant. If somebody could explain to me how to interpret this problem, I think that's really all I'm missing here. I just need help getting an equation to differentiate, as well as help getting the proportionality constant.

Thank you!

2. The first one is fairly easy. The problem says " The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed v.". Letting "C" be the cost of fuel in dollars per hour and "v" the speed in miles per hour, then $\displaystyle C= kv^3$ (that's what "proportional" means).

Since you are also told "A certain ferry boat uses $196 worth of fuel per hour when cruising at 10 miles per hour", put C= 196 and v= 10 in that equation and solve for k. Going at speed "v miles per hour", it will take, of course, 1/v hours to go one mile. Since the cost of fuel is$\displaystyle kv^3$dollars per hour, the fuel cost to go 1 mile is$\displaystyle (kv^3)(1/v)= kv^2$dollars. Since the cost of everything else is 1323 dollars per hour, that cost, to go one mile, will be 1323/v dollars. The total cost will be the sum of those:$\displaystyle kv^3+ 1323/v\$