✔ 最佳答案
a. Now, consider the cardboard: xy = 3600
y = 3600 / x
Consider the picture:
(y - 12 - 13)(x - 8 - 8) = A
(y - 25)(x - 16) = A
xy - 25x - 16y + 400 = A
3600 - 25x - 16(3600 / x) + 400 = A
A = -25x + 4000 - 57 600/x
b. dA/dx = -25 + 57 600/x2
d2A/dx2 = -115 200/x3
Set dA/dx = 0
-25 + 57 600/x2 = 0
x = 48 or -48 (rejected) (Since x > 0)
When x = 48, d2A/dx2 < 0
So, max. of A is obtained when x = 48
So, the larger value of A
= -25(48) + 4000 - 57 600/48
= 1600
c.i. As A attains a maximum when x = 48.
And there is only one turning point.
So, for x > 48, A decreases as x increases.
ii. For x >= 50, A decreases as x increases.
So, A is largest when x = 50
that is, max. of A
= -25(50) + 4000 - 57 600/50
= 1598
d. 4/9 < x/y < 9/16
4y/9 < x < 9y/16
4/9 (3600/x) < x < 9/16 (3600/x)
1600/x < x < 2025/x
1600 < x2 < 2025
40 < x < 45 (Since x > 0)
For this region, A increases as x increases.
So, the largest value of A is obtained when x = 45
Largest value of A
= -25(45) + 4000 - 57 600/45
= 1595