Question

find the volume of each regular prism. leave answer in exact form.
2)
3)
base area =
height =
volume =
base area =
height =
volume =

Explanation:

Step1: Calculate base - area of the first prism

The base of the first prism is a regular pentagon. The area formula for a regular pentagon is $A=\frac{1}{2}\times perimeter\times apothem$. The perimeter of the pentagon with side - length $s = 10$ m is $P = 5s=5\times10 = 50$ m, and the apothem $a = 6.88$ m. So the base - area $A_1=\frac{1}{2}\times50\times6.88=172$ m².

Step2: Find volume of the first prism

The volume formula of a prism is $V = A\times h$, where $A$ is the base - area and $h$ is the height. Given $h_1 = 14$ m and $A_1 = 172$ m², then $V_1=172\times14 = 2408$ m³.

Step3: Calculate base - area of the second prism

The base of the second prism is a regular hexagon. The area formula for a regular hexagon can be divided into six equilateral triangles. If the side - length of the hexagon $s = 5$ in, the area of a single equilateral triangle with side - length $s$ is $A_{\triangle}=\frac{\sqrt{3}}{4}s^{2}$. The base - area of the hexagon $A_2 = 6\times\frac{\sqrt{3}}{4}s^{2}=6\times\frac{\sqrt{3}}{4}\times5^{2}=\frac{75\sqrt{3}}{2}$ in².

Step4: Find volume of the second prism

Given the height of the second prism $h_2 = 8$ in and $A_2=\frac{75\sqrt{3}}{2}$ in², then $V_2=A_2\times h_2=\frac{75\sqrt{3}}{2}\times8 = 300\sqrt{3}$ in³.

Answer:

For the first prism: BASE AREA = 172 m², HEIGHT = 14 m, VOLUME = 2408 m³
For the second prism: BASE AREA = $\frac{75\sqrt{3}}{2}$ in², HEIGHT = 8 in, VOLUME = $300\sqrt{3}$ in³