The base of a triangular prism is a right triangle with hypotenuse 10 m long and one leg 6 m long. If the height of the prism is 12 m, what is the...

Saturday, June 29, 2013

The base of a triangular prism is a right triangle with hypotenuse 10 m long and one leg 6 m long. If the height of the prism is 12 m, what is the...

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For a prism its volume $\displaystyle{V}$ is $\displaystyle{S}\cdot{H},$ where $\displaystyle{S}$ is the area of a base and $\displaystyle{H}$ is the height of a prism. $\displaystyle{H}={12}{m}$ is given so it is sufficient to find $\displaystyle{S}.$

A base is a right triangle, its area may be computed as $\displaystyle\frac{{1}}{{2}}{L}_{{1}}\cdot{L}_{{2}},$ where $\displaystyle{L}_{{1}}$ and $\displaystyle{L}_{{2}}$ are the lengths of the legs. Also denote the length of the hypotenuse as $\displaystyle{G}.$

One leg, $\displaystyle{L}_{{1}}={6}{m},$ is given, and we can find the second by the Pythagorean theorem: $\displaystyle{L}_{{2}}=\sqrt{{{{G}}^{{2}}-{{L}_{{1}}^{{2}}}}}=\sqrt{{{{10}}^{{2}}-{{6}}^{{2}}}}=\sqrt{{{64}}}={8}{\left({m}\right)}.$

Thus the formula is $\displaystyle{V}=\frac{{1}}{{2}}{H}\cdot{L}_{{1}}\cdot\sqrt{{{{G}}^{{2}}-{{L}_{{1}}^{{2}}}}}.$ In numbers it is $\displaystyle\frac{{1}}{{2}}\cdot{12}\cdot{6}\cdot{8}={288}{\left({{m}}^{{3}}\right)}.$ This is the answer.

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Saturday, June 29, 2013