Page 129 - How To Solve Word Problems In Calculus

relative maximum. Since it is the only relative extremum on (0, ∞),
it gives the absolute maximum area.
8.

y
x
x

2
V = x y. The postal restrictions say that 4x + y cannot exceed
108 inches. For maximum volume we take 4x + y = 108.
Equivalently, y = 108 − 4x and 0 ≤ x ≤ 27. (If x > 27, y < 0.)
2
V (x) = x (108 − 4x)
2
= 108x − 4x 3
V (x) = 216x − 12x 2

0 = 216x − 12x 2
0 = 12x(18 − x)

The critical values are x = 0 and x = 18. Since the endpoints of the
interval, x = 0 and x = 27, both yield a volume of 0, the maximum
volume occurs when x = 18 and its value is

2
V max = V (18) = 18 (108 − 4 · 18) = 324 · 36 = 11,664 in 3
9.

E

13
5 C

y
x 12 − x
A
D B
12

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