minimisation question (1 Viewer)

[skillstriker]

A traveller employs a man to drive him from Sydney to Wollongong for an hourly payment of P dollars. Running costs of the car, which are also paid by the traveller, are kv³ dollars per hour, where v km/h is the speed, and k is a constant. Find the uniform speed that will minimise the total cost of the journey.

Please show full working. Thanks.

 

[bleakarcher]

C=(P+kv^3)t
t=d/v
=> C(v)=(P+kv^3)(d/v)=Pd/v + kdv^2
C '(v)=-Pd/v^2 + 2kdv
To minimise C we set C '(v)=0,
=> 2kdv^3=Pd
v^3=P/2k
v=cuberoot(P/2k)
show that its a minimum