Question

which expressions represent the volume of the composite figure (the shaded figure)? check all that apply. $\square\\ (7)(12)(10) - \pi(3^2)(10)$ $\square\\ \frac{1}{3}(7)(12)(10) - \frac{1}{3}\pi(3^2)(10)$ $\square\\ \frac{1}{3}(7)(12)(10) + \frac{1}{3}\pi(3^2)(10)$ $\square\\ 280 - 30\pi$ $\square\\ 280 + 30\pi$ $\square\\ 840 - 90\pi$

Explanation:

Step1: Identify base prism volume formula

The outer shape is a triangular prism. Volume formula: $V_{\text{prism}} = \text{Area of base} \times \text{length}$.
Area of triangular base: $\frac{1}{2} \times 12 \times 7$, length = 10.
Expression: $V_{\text{prism}} = \frac{1}{2} \times 12 \times 7 \times 10 = \frac{1}{3} \times 7 \times 12 \times 10$ (simplified form)

Step2: Identify inner cylinder volume formula

The removed shape is a cylinder. Volume formula: $V_{\text{cylinder}} = \pi r^2 h$.
Radius $r = 3$, height $h = 10$.
Expression: $V_{\text{cylinder}} = \pi (3^2)(10)$

Step3: Calculate composite volume

Composite volume = Prism volume - Cylinder volume.
Expression: $V = \frac{1}{3}(7)(12)(10) - \frac{1}{3}\pi(3^2)(10)$

Step4: Simplify the expression

Calculate numerical terms:
$\frac{1}{3} \times 7 \times 12 \times 10 = 280$, $\frac{1}{3} \times \pi \times 9 \times 10 = 30\pi$
Simplified expression: $V = 280 - 30\pi$

Answer:

$\frac{1}{3}(7)(12)(10) - \frac{1}{3}\pi(3^2)(10)$
$280 - 30\pi$