2000 HSC Question 7 a ii (1 Viewer)

[DVDVDVDV]

I did a past paper today, 100% except for this question. I loooked at the solutions and I still don't get it. Part i) is fine. Part ii) screws me up big time. Can someone help?

"The amount of fuel F in litres required per hour to propel a plane in level flight at constant speed u km/h is given by

F = Au^3 + B/u

where A and B are constants."

i) "Show that a pilot wishing to remain in level flight for as long a period as possible should fly at

(B/3A)^0.25 km/h"

ii) "Show that a pilot wishing to fly as far as possible in level flight should fly approximately 32% faster than the speed given in part (i)"

Could someone solve this for me and explain the steps for me please?

 

[nightweaver066]

i)














ii)




















%

 

[DVDVDVDV]

I did say the first bit wasn't a problem...

 

[math man]

for max distance we need to incorporate distance into this expression using:

distance = speed(u) x time

now we know F is the rate of fuel per hour in litres, therefore:

which reads fuel is rate of change of litres with respect to time. If we invert and integrate we find that:

the constant goes to 0 as t= 0 means l =0.

now we have:



subbing in F and simplyfying gives:



differentiating gives:




setting to zero to find max/min gives:



which gives:



and we get:



now if we compare this speed to the original speed from i):




we get the ratio approx equal to:



which is approx 1.32, therefore the new speed is 32% greater