Question

writing expressions for volume of a composite figure
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 solid volume

The composite figure is a triangular prism with a cone removed. First, calculate the volume of the triangular prism:
Volume of triangular prism = $\frac{1}{2} \times \text{base} \times \text{height} \times \text{length} = \frac{1}{3} \times 7 \times 12 \times 10$

Step2: Identify removed solid volume

Next, calculate the volume of the cone (to subtract):
Volume of cone = $\frac{1}{3} \times \pi \times r^2 \times h = \frac{1}{3} \pi (3^2)(10)$

Step3: Write composite volume expression

Volume of shaded figure = Volume of prism - Volume of cone:
$\frac{1}{3}(7)(12)(10) - \frac{1}{3}\pi(3^2)(10)$

Step4: Simplify the expression

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

Answer:

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