The horizontal base of a solid prism is an equilateral triangle of side x the sides of the prism are vertical the height of prism is h and volume=2000 Express h in term of x show that total surface area A =(3^1/2)/2*x^2 +(24000/3^1/2)x^-1?

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The horizontal base of a solid prism is an equilateral triangle of side x the sides of the prism are vertical the height of prism is h and volume=2000 Express h in term of x show that total surface area A =(3^1/2)/2*x^2 +(24000/3^1/2)x^-1?

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Answer 1 · 2018-05-08T15:35:27

Volume for the prism is Base Area times the Height.
So we substitute our values in the equation as follows;
$V = (\frac{1}{2} \cdot x \cdot x \cdot sin 60)$ $V = (1/2 * x * x * sin60)$

$2000 = (\frac{1}{2} \cdot x^{2} \cdot sin 60)$ $2000 = (1/2 * x^2 * sin60)$

$2000 = \frac{\sqrt{3}}{4} . x^{2} h$ $2000 = sqrt(3)/4 . x^2 h$

$\frac{8000}{\sqrt{3}} . x^{2} = h$ $8000/sqrt(3). x^2 = h$

Now that we have our $h$ $h$ let's find Surface Area;
$A = 3 (h . x) + 2 (\frac{1}{2} . a . b . sin 60)$ $A = 3(h.x) + 2(1/2 .a. b. sin 60)$

$= 3 h x + x^{2} sin 60$ $= 3hx + x^2sin60$

$3 (\frac{8000}{\sqrt{3} x^{2}})$ $3(8000/(sqrt(3)x^2))$ . $x + \frac{\sqrt{3}}{2} . x^{2}$ $x + sqrt(3)/2.x^2$

$\frac{24000}{\sqrt{3}} x^{- 1} + \frac{\sqrt{3}}{2} x^{2}$ $24000/sqrt(3)x^-1 + sqrt(3)/2 x^2$

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The horizontal base of a solid prism is an equilateral triangle of side x the sides of the prism are vertical the height of prism is h and volume=2000 Express h in term of x show that total surface area A =(3^1/2)/2*x^2 +(24000/3^1/2)x^-1?

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