Speed Boat Maintenance
Walk me through this calculus 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. A certain ferry boat uses $100 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 $675 per hour. At what speed should it travel so as to minimize the cost per mile traveled?”
I’m a tad lost on how to find the formulas that I’ll eventually take the derivitive of, set equal to 0 and so forth… :/
I know the answer is 15 mph, by the way. I just can’t arrive there. 
f(v) = kv^3
100 = 10^3 * k = 1000k
k = 0.1
cost per hour, C(v)
= 0.1v^3 + 675
cost per mile, C(v) / v
= 0.1v^2 + 675/v
[C(v) / v]‘
= 0.2v – 675/v^2 = 0
v^3 = 675 / 0.2 = 3375
v = 15 mph
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