急 ! F4 Quadratic Functions

23.
L = 9x + 8y => y = (L-9x)/8
A = (3x)(2y) = 6xy => (-27x^2 + 3Lx)/4......(1)
23a.
d(A)/dx = A' = (-54x + 3L)/4
d(A')/dx = A'' = -27/2 < 0 => A will be maximised
Max A when A'=0
=> -54x + 3L = 0 => x = L/18
thus, sub x = L/18 into (1) => max A = L^2 / 48
23b.
max A => x = L/18 => y = L/16
x/y = 8/9

26.
26a.
Quautity, Q = 200 + (10x/0.1) = 200 + 100x
Profit, P = (5-x-2) Q => P = (3-x)(200+100x)
P = -100x^2 + 100x + 600
26b.
d(P)/dx = P' = -200x + 100x
d(P')/dx = P'' = -200 < 0 => P will be maximised
Max P when P'=0
=> -200x + 100 = 0 => x = 0.5 => selling price = 5-x = 4.5
P = (2.5)(250) = 625