a.maths differentiation applications 2

bscat2005

2008-05-15 9:09 am

A solid triangular prism has a given volume of V cm^3 . its base is an equilateral triangle .

Find the length of each side of the equilateral triangle so that the total surface area of the prism is a minimum

回答 (1)

用戶10012

2008-05-16 1:11 am

✔ 最佳答案

Let x be the side of the equilateral triangle. Therefore, area of the equilateral triangle =sqrt3(x^2)/4.
Let height of the prism be h. Then volume of the prism=hsqrt3(x^2)/4=V or h=4V/sqrt3(x^2).
Let Surface area of the prism be S. Then S=3xh + 2[sqrt3(x^2)/4].
Substitute h into the equation, S=12xV/sqrt3(x^2) + sqrt3(x^2)/2. Simplify it to S=12V/sqrt3(x) +sqrt3(x^2)/2.
Therefore, dS/dx = -12V/sqrt3(x^2) + sqrt3(x).
Put dS/dx =0, we get -12V/sqrt3(x^3) + sqrt3 =0,
that is x^3 = 4V, therefore, x=(4V)^1/3.

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