Question

select the correct answer.
a rectangular prism has a length of 7 feet, a width of 4 feet, and a height of 9 feet. what is the approximate radius of a sphere with the same surface area as the rectangular prism?
4.5 ft
3.2 ft
10.1 ft
6.4 ft

Explanation:

Step1: Calculate prism surface area

The surface area formula for a rectangular prism is $SA_{prism}=2(lw + lh + wh)$, where $l=7$, $w=4$, $h=9$.
$SA_{prism}=2(7\times4 + 7\times9 + 4\times9)=2(28 + 63 + 36)=2\times127=254$ square feet

Step2: Set sphere area equal to prism area

The surface area formula for a sphere is $SA_{sphere}=4\pi r^2$. Set equal to 254:
$4\pi r^2=254$

Step3: Solve for $r$

First isolate $r^2$: $r^2=\frac{254}{4\pi}=\frac{127}{2\pi}\approx\frac{127}{6.283}\approx20.21$
Then take square root: $r\approx\sqrt{20.21}\approx4.5$

Answer:

4.5 ft