Question

What is the volume of the composite figure? Explain your work. A complete answer should include how you broke up the figure, which numbers you multiplied to find the volume. You may want to use the formula for volume to find the solution.

Answer 1

`15,000\:\mathrm{mm^3}`

Explanation:

The composite figure consists of a square prism and a trapezoidal prism. By adding the volume of each, we obtain the volume of the composite figure.

The volume of the square prism is given by
`V=s^2\cdot h`, where
`s` is the base length and
`h` is the height. Substituting given values, we have:
`V=14^2\cdot 30=196\cdot 30=5,880\:\mathrm{mm^3}`

The volume of a trapezoidal prism is given by
`V=(b_1+b_2)/(2)\cdot l\cdot h`, where
`b_1` and
`b_2` are bases of the trapezoid,
`l` is the length of the height of the trapezoid and
`h` is the height. This may look very confusing, but to break it down, we're finding the area of the trapezoid (base) and multiplying it by the height. The area of a trapezoid is given by the average of the bases (
`(b_1+b_2)/(2)`) multiplied by the trapezoid's height (
`l`).

Substituting given values, we get:

`V=(14+24)/(2)\cdot (30-14)\cdot 30,\\V=19\cdot 16\cdot 30=9,120\:\mathrm{mm^3}}`

Therefore, the total volume of the composite figure is
`5,880+9,120=\boxed{15,000\:\mathrm{mm^3}}` (ah, perfect)

Alternatively, we can break the figure into a larger square prism and a triangular prism to verify the same answer:

`V=30^2\cdot 14+(1)/(2)\cdot10\cdot 16\cdot 30=\boxed{15,000\:\mathrm{mm^3}}\checkmark`