Optimisation - Chain Rule

Question

So far, the only way I could think to do this is to use the chain rule by splitting up the cost function: dC/dt = dC/dx * dx/dt

But, im not sure what I could use as dx/dt as there is no function for this. Any help is appreciated Thankyou!!

Answer 1

Write total cost $C$ as $$C=\frac{v^3}{10}t+675t$$ Assuming it travels at a constant speed, $v=x/t$. Here $x$ is the total distance travelled, $t$ is the total time of the journey. You are asked to minimise cost per unit of distance travelled. So minimise $C/x$. We have $t=x/v$. Then $$C=\frac{v^2}{10}x+\frac{675}vx\\\implies C/x=\frac{v^2}{10}+\frac{675}{v}$$

Then to find the minimum, solve $$\frac{d(C/x)}{dv}=0.$$

This should give $v=15$.

Answer 2

$C/hr = V^3/10 + 675$

$C/km = 1/V*(V^3/10 +675)$

$C/km = V^2/10 + 675/V$

$d(C/km)/dV = 2V/10 - 675/V^2$

$d(C/km)/dV$ is a minimum when......

$2V/10 - 675/V^2 = 0$

$675/V^2 = 2V/10$

$2V^3 = 6750$

$V = 15 km/hr$