Question

Using the Triangular Prism in the picture, find the Lateral Area, the Area of a Single Base, and the TOTAL Surface Area7 ft?? ft45 ftFirst find the missing length =Lateral Area =ft²Single Base Area =Surface Area =Blank 1:Blank 2:ft²ft2ft

Answer 1

Explanation

We are given the following:

`\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}`

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:

`\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=√(9)=3ft \end{gathered}`

Hence, the missing length of the triangle (base) is 3ft.

Next, the lateral area can be determined as:

`\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((1)/(2)bh)+(lw)+(lw) \\ Area=2((1)/(2)*3*4)+(7*3)+(7*4) \\ Area=2(6)+21+28 \\ Area=12+21+28=61\text{ }square\text{ }ft. \end{gathered}`

Hence, the lateral area is 61 square feet.

The base area can be calculated thus:

`\begin{gathered} Area=lw \\ Area=7*5 \\ Area=35\text{ }square\text{ }feet \end{gathered}`

Hence, the single base area is 35 square feet.

Finally, we can calculate the surface area of the figure as:

`\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}`

Hence, the surface area of the figure is 96 square feet.