Question

Find the volume and surface area please!! please explain step by step so I understand.

Answer 1

In order to calculate the surface area, first let's calculate the area of the pentagon that is the base of the prism.

To do so, let's divide it in a rectangle and triangle. The rectangle has dimensions 15 ft by 9 ft, and the triangle has base 15 ft and other two sides of 8 ft.

The area of the rectangle is given by:

`A_r=15\cdot9=135`

And the area of the triangle can be calculated with Heron's formula:

`\begin{gathered} A_t=\sqrt[]{p(p-a)(p-b)(p-c)} \\ p=(15+8+8)/(2)=15.5 \\ A_t=\sqrt[]{15.5(15.5-15)(15.5-8)(15.5-8)} \\ A_t=20.88 \end{gathered}`

So the pentagon area is:

`A_p=A_r+A_t=135+20.88=155.88`

The areas of each lateral rectangular face is:

`\begin{gathered} A_1=15\cdot30=450 \\ A_2=A_3=30\cdot9=270 \\ A_4=A_5=30\cdot8=240 \end{gathered}`

So the surface area of the prism is:

`\begin{gathered} S=2\cdot A_p+A_1+A_2+A_3+A_4+A_5 \\ S=311.76+450+270+270+240+240 \\ S=1781.76 \end{gathered}`

The volume can be calculated as the base area multiplied by the height of 30 ft:

`\begin{gathered} V=A_p\cdot h \\ V=155.88\cdot30 \\ V=4676.4 \end{gathered}`

The amount of glass needed to cover the greenhouse is:

`\begin{gathered} A=A_2+A_3+A_4+A_5 \\ A=270+270+240+240 \\ A=1020 \end{gathered}`

So the amount of glass needed is 1020 ft².