MATH!!!!!!VERY 簡單(calculus!!!

tze

2011-07-29 2:05 am

a rectangular box is made such that its lengh is twice its width and the total lengh of its 12 edges is 180cm,find the dimensions of the box which has the maximum volume
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回答 (1)

不用客氣

2011-07-29 6:59 am

✔ 最佳答案

Lety cm be the width of the rectangle.
Then, the length of the rectangle = 2y cm
and the height of the rectangle = [180 - 4y - 4(2y)]/4 cm = (45 - 3y) cm

Denote V cm³ as the volume of the rectangle.
Then V = y * (2y) * (45 - 3y) = 90y² - 6y³
dV/dy = 180y - 18y² = 180y(10 - y)
d²V/dy² = 180 - 36y

y ≠ 0
When y = 10
dV/dy = 0 and d²V/dy² = -180 < 0
Hence, When V is maximum, y = 10
2y = 20
(45 - 3y) = 15

When the box has the maximum volume,
dimension of the box: 20 cm x 10 cm x15 cm

參考: micatkie

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