You are returning home following a summer break in a place where the temperature

Question

You are returning home following a summer break in a place where the
temperature is around 70 F. You will fly your personal Beachcraft King/Queen Air.
You want to choose a cruise altitude to minimize your total flight time. Here is the data you need to make you decision.
-You are flying a total distance L=1820 km.
-The King/Queen air can climb at a rate dh / dt=540 m/min
-The groundspeed during the climb is Vx=240km/hour
-You plan a descent at dh / dt=500 m/min
-Your ground speed during descent will be Vx=460 km/hour
-Your cruise speed of the king/Queen Air varies with altitude as:
Vx=415+0.0291h-1.99*10-6h2
for Vx in km/hour and h in m.
What is the altitude you want to cruise at and how long will it take you to get home? (hint: use secant method if taking the derivative is too messy)

Explanation / Answer

Given, Velocity (in cruise) Vx=415+0.0291h-1.99*10-6h2km/hr

for finding max cruise velocity, Vmax we need to the local maxima of the above function

i.e. dVx /dh = 0

0.0291 - 2*1.99*10-6h = 0

Cruise Height h = 0.0291/[2*1.99*10-6] = 7311.557 meters (Answer)

d2Vx /dh2 = -2*1.99*10-6 (Negative) So, we have found the local maxima.

Max. Cruise velocity Vx = 415+(0.0291*7311.557)-1.99*(7311.557)2*10-6

Vx = 521 km/hr

Time of Climb = h/[dh / dt]climb = 7311.557/ 540 = 13.54 mins

Ground distance covered during Climb = Vx climb * Tclimb = 240* 13.54/60 = 54.16 km

Time of Descent = h/[dh / dt]descent= 7311.557/ 500 = 14.623 mins

Ground distance covered during Descent = Vx descent * Tdescent = 460* 14.623/60 = 112.11 km

Remaining distance to be covered in cruise = 1820 - 54.16 - 112.11 = 1653.73 km

Time for Cruise = 1653.73 / 521 = 3.17 hrs

Total time = 3.17 + (13.54/60) + (14.623/60) = 3.64 hrs (Answer)

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