請仔細回答例明詳細 THX!!

2012-10-04 8:49 pm

Find the dimensions of the rectangular box with a square base and no top that has a volume of 50000 cubic centimeters and the smallest possible surface area.
Round your answer to the nearest hundredth of centimeter.

回答 (1)

micatkie

2012-10-05 1:53 am

✔ 最佳答案

Let x cm³ be the length of each sideof the square base.
Then, the height = 50000/x² cm

Denote the surface area as A cm².
A = x² + 4x(50000/x²)
A = x² + (200000/x)

dA/dx
= 2x - (200000/x²)
= (2x³ - 200000)/x²
= 2(x³ - 100000)/x²

d²A/dx² = 2 + (100000/x³)

When wx = ³√100000 :
x³ = 100000
dA/dx = 2(100000 - 100000)/(³√100000)² = 0
d²A/dx² = 2 + (100000/1000000) = 3> 0

Hence, when wx = ³√100000:
Minimum Area
= (³√100000)² + (200000/³√100000) cm²
= 6463.30 cm²

參考: micatkie

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請仔細回答 例明詳細 THX!!

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請仔細回答例明詳細 THX!!