Question

A right rectangular prism measures 8 inches long, 12 inches high, and 6 inches deep. A half-sphere with a diameter of 2 inches is carved out of the prism. What is the approximate volume of the resulting composite figure?

Answer 1

Answer:

$573.9 \text{ in}^3$

Step-by-step explanation:

First, we can find the volume of the rectangular prism using the formula:

$V_\square = l \cdot w \cdot d$

where $l$ is the prism's length, $w$ is its width, and $d$ is its depth.

Plugging the given dimensions into the formula:

$V_\square = 8 \cdot 12 \cdot 6$

$V _\square= 96 \cdot 6$

$\boxed{V_\square = 576 \text{ in}^3}$

Next, we can find the volume of the half-sphere using the formula:

$V_\circ = \dfrac{2}{3} \pi r^3$

where $r$ (or $d/2$) is the half-sphere's radius.

Plugging the given diameter value into the formula:

$V_\circ = \dfrac{2}{3} \pi (2/2)^3$

$V_\circ = \dfrac{2}{3} \pi (1)^3$

$\boxed{V_\circ=\dfrac{2}{3}\pi \text{ in}^3}$

Finally, we can find the volume of the composite figure by subtracting the volume of the half-sphere from the volume of the rectangular prism.

$V = V_\square - V_\circ$

$V = 576 \text{ in}^3 - \dfrac{2}{3}\pi \text{ in}^3$

We can evaluate this using a calculator.

$\boxed{V\approx 573.9 \text{ in}^3}$