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...

Hello!

For a prism its volume `V` is `S*H,` where `S` is the area of a base and `H` is the height of a prism. `H=12 m` is given so it is sufficient to find `S.`

A base is a right triangle, its area may be computed as `1/2 L_1*L_2,` where `L_1` and `L_2` are the lengths of the legs. Also denote the length of the hypotenuse as `G.`

One leg, `L_1=6 m,` is given, and we can find the second by the Pythagorean theorem: `L_2 = sqrt(G^2 - L_1^2) = sqrt(10^2-6^2) = sqrt(64) = 8 (m).`

Thus the formula is `V=1/2 H*L_1*sqrt(G^2 - L_1^2).` In numbers it is `1/2*12*6*8 = 288 (m^3).` This is the answer.