急 ! F5 Variations 1條 7q7

(a)
Let P = (k1)x + (k2)x^2 , where k1, k2 are constants.
then
12500 = 50k1 + 2500k2 ... (i)
and
20000 = 100k1 + 10000k2 ... (ii)

(ii) - 2x(i):
-5000 = 5000k2
k2 = -1

Put k2 = -1 into (i):
12500 = 50k1 - 2500
k1 = 300

i.e. P = 300x - x^2

(b)
When P > 12500
300x - x^2 > 12500
x^2 - 300x + 12500 < 0
(x - 50)(x - 250) < 0

(x - 50 > 0 and x - 250 < 0) OR (x - 50 < 0 and x - 250 > 0)
(x > 50 and x < 250) OR (x < 50 and x > 250) (rejected)
i.e. Required answer: 50 < x < 250.

(c)
P = 300x - x^2
= -(x^2 - 300x)
= -[(x - 150)^2 - 150^2]
= -(x - 150)^2 + 22500
i.e. The greatest profit = $22500