NCEA Calculus 2014 TWO(d - Solution & Explanation

Question

The hourly cost of running an aeroplane depends on the speed at which it flies.

For a particular aeroplane this is given by the equation

C = 4v + \dfrac{1\ 000\ 000}{v},\ 200 \le v \le 800

where

C

is the hourly cost of running the aeroplane, in dollars per hour
and

v

is the airspeed of the aeroplane, in kilometres per hour.

Find the minimum hourly cost at which this aeroplane can be flown.

*Show any derivatives that you need to find when solving this problem.*

Official Answer

C = 4v + \dfrac{1000000}{v}

\dfrac{dC}{dv} = 4 - \dfrac{1000000}{v^2}

Minimum when

\dfrac{dC}{dv} = 0

v^2 = 250000

v = 500

C = 4 \times 500 + \dfrac{1000000}{500} = 4000

Grading Criteria

Achievement (u)

Correct value for $v$ with correct derivative.

Merit (r)

A correct solution.
Units not required.

Excellence T1

-

Excellence T2

-

2014 TWO(d)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2