Explanation
We are given the following:
[tex]\begin{gathered} Lateral\text{ }triangle:\begin{cases}height={4ft} \\ hypotenuse={5ft}\end{cases} \\ \\ Right\text{ }Lateral\text{ }rectangle:\begin{cases}length={7ft} \\ width={4ft}\end{cases} \\ \\ Left\text{ }Lateral\text{ }rectangle:\begin{cases}length={7ft} \\ width={?}\end{cases} \\ \\ Base\text{ }rectangle:\begin{cases}length={7ft} \\ width={5ft}\end{cases} \end{gathered}[/tex]We are required to determine the following:
• The missing length of the lateral triangle.
,• The lateral area.
,• The single base area.
,• The surface area of the figure.
We can obtain the missing length of the lateral triangle by using the Pythagorean theorem as follows:
[tex]\begin{gathered} Hyp.^2=Opp.^2+Adj.^2 \\ Hyp.^2=Height^2+Base^2 \\ 5^2=4^2+Base^2 \\ Base^2=5^2-4^2 \\ Base^2=9 \\ Base=\sqrt{9}=3ft \end{gathered}[/tex]Hence, the missing length of the triangle (base) is 3ft.
Next, the lateral area can be determined as:
[tex]\begin{gathered} Area=Area\text{ }of\text{ }lateral\text{ }triangles+Area\text{ }of\text{ }lateral\text{ }rectangles \\ There\text{ }are\text{ }two\text{ }lateral\text{ }triangles\text{ }(front\text{ }and\text{ }back)\text{ }and\text{ }two\text{ }lateral\text{ }rectangles\text{ }(left\text{ }and\text{ }right) \\ \therefore Area=2(\frac{1}{2}bh)+(lw)+(lw) \\ Area=2(\frac{1}{2}\times3\times4)+(7\times3)+(7\times4) \\ Area=2(6)+21+28 \\ Area=12+21+28=61\text{ }square\text{ }ft. \end{gathered}[/tex]Hence, the lateral area is 61 square feet.
The base area can be calculated thus:
[tex]\begin{gathered} Area=lw \\ Area=7\times5 \\ Area=35\text{ }square\text{ }feet \end{gathered}[/tex]Hence, the single base area is 35 square feet.
Finally, we can calculate the surface area of the figure as:
[tex]\begin{gathered} Surface\text{ }Area=Area\text{ }of\text{ }lateral\text{ }faces+Area\text{ }of\text{ }base \\ Surface\text{ }Area=61+35 \\ Surface\text{ }Area=96\text{ }square\text{ }feet \end{gathered}[/tex]Hence, the surface area of the figure is 96 square feet.
