Question

type the correct answer in each box. use numerals instead of words. a triangular prism has an approximate surface area of 39.5 square feet. the sides of the equilateral triangular bases are one - third the length of the prism. what are the dimensions of the prism to the nearest foot? side of triangular base ≈ ft length of prism ≈ ft

Explanation:

Step1: Recall surface - area formula

The surface - area formula for a triangular prism with equilateral triangular bases is $SA = 2\times\frac{\sqrt{3}}{4}s^{2}+3sl$, where $s$ is the side length of the equilateral - triangular base and $l$ is the length of the prism. Given that $l = 3s$.
Substitute $l = 3s$ into the surface - area formula:
$SA=2\times\frac{\sqrt{3}}{4}s^{2}+3s\times(3s)$
$SA=\frac{\sqrt{3}}{2}s^{2}+9s^{2}$
$SA = s^{2}(\frac{\sqrt{3}}{2}+9)$

Step2: Solve for $s$

We know that $SA = 39.5$. So,
$39.5=s^{2}(\frac{\sqrt{3}}{2}+9)$
First, calculate $\frac{\sqrt{3}}{2}+9\approx\frac{1.732}{2}+9 = 0.866+9=9.866$
Then, $s^{2}=\frac{39.5}{9.866}\approx4$
$s=\sqrt{4}=2$ (rounded to the nearest foot)

Step3: Solve for $l$

Since $l = 3s$, when $s = 2$, $l=3\times2 = 6$

Answer:

side of triangular base $\approx2$ ft
length of prism $\approx6$ ft