Maths Question help - Optimisation?

Question

The cost of running a train at a constant speed v km/h is

C = 50 + (v^2/1000) dollars/h.

Find the most economical speed for a journey of 800km.

the answer is approximately 223.60km/h - i just don't know how to get here. please help :)

Answer 1

Graph it using C as your y (axis) and v as your x and all will become clear.

Answer 2

t = 800 / v

cost, C(v)

= [50 + (v^2/1000)] * t

= [50 + (v^2/1000)] * (800 / v)

= 40000/v + 4v/5

C'(v) = -40000/v^2 + 4/5 = 0

v^2 = 40000 * 5/4 = 50000

v = √50000 = 223.60679775 (only the +ve answer accepted).