Simple optimization problem solving

Question

A small boat moving at $x$ km/h uses fuel at a rate that is approximated by the function$$q=8+\dfrac{x^2}{50}$$ where $q$ is measured in litres/h.

Determine the speed of the boat for which the amount of fuel used for any given journey is least.

In attempting to answer this question, I assumed that you had to find when $q$ was a minimum. I found the derivative of the function, and made it equal to $0$, however this gets $x=0$. The answer is $x=20$. I am sure I'm missing something obvious, like multiplying by the number of hours, but I tried this and still cant seem to get the right answer.

Answer 1

Let minimize the fuel used per km that is that is

• fuel consuming in time T: qT
• distance in time T: xT

then

• fuel used per km $c(x)=\frac{qT}{xT}=\frac{q}{x}=\frac{8}{x}+\dfrac{x}{50}\implies c'(x)=-\frac{8}{x^2}+\frac1{50}=0\implies x=20$

Answer 2

Let's say the journey takes $t$ hours. Then the distance the boat passes is $d=xt$ or $t=\frac{d}{x}$. In $t$ hours total $qt$, that is $qt=\frac{qd}{x}$ fuel will be used. To find the least fuel per journey you must minimize: $$\frac{\text{total fuel}}{\text{distance}}=\frac{\frac{qd}{x}}{d}=\frac{q}{x}=\frac{8}{x}+\frac{x}{50} \Rightarrow \left(\frac{q}{x}\right)'=-\frac{8}{x^2}+\frac{1}{50}=0 \Rightarrow x=20.$$