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In a regular triangular prism the distance from the centre of one base to one of the vertices of the otherbase is l . The altitude of the prism for which the volume is greatest

In a regular triangular prism the distance from the centre of one base to one of the vertices of the otherbase is l . The altitude of the prism for which the volume is greatest

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
Ans:
Let the altitude of th prism to be ‘h’ & height of triangle part of the prism to be ‘H’.
Given:
(\frac{H}{2})^{2}+(\frac{h}{2})^{2} = l^{2}
H = 2\sqrt{l^{2}-(\frac{h}{2})^{2}}
Area of triangle:
A = \frac{H^{3}}{\sqrt{3}} = \frac{4}{\sqrt{3}}(l^{2}-\frac{h^{2}}{4})
Volume of Prism ‘V’ :
V = \frac{1}{2}Ah = \frac{4h}{\sqrt{3}}(l^{2}-\frac{h^{2}}{4})
To maximise the volume,
\frac{\partial V}{\partial h} = \frac{4}{\sqrt{3}}(l^{2}-\frac{3h^{2}}{4}) = 0
\Rightarrow h = \frac{2l}{\sqrt{3}}
Thanks & Regards
Jitender Singh
IIT Delhi
askIItians Faculty

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