The operating cost, C, in dollars per hour, for an airplane cruising at a height of h metres and an air speed of 200km/h is given by:

Question

C = 4000 + h/15 + 15000000/h

for the domain 1000 ≤ h ≤ 20000.

Determine the height at which the operating cost is at a minimum, and find the operating cost per hour at this height.

So far this is what I've tried:

C' = 4000 + 1/15 -15000000/h^2

I have no clue what to do next lol plz help

Answer 1

1. find derivative with respect to h:
C'(h) = 1/15-15000000/h²
2. equate C'(h) to zero and solve for h.
2. 1/15-15000000/h² = 0
h²=225000000
h=15000
3. verify that it is a minimum (not maximum) by taking second derivative, and see that C"(15000)>0.
(if C"(15000)<0, then it is a maximum).
C"(h)=30000000/h³
C"(15000)=1/112500 > 0
so h=15000 is a minimum.