Question

find the surface area and volume of the solid. round each measure to the nearest tenth, if necessary. volume: 800 ft³ surface area: 1280 ft² need help with this question? incorrect 2 tries left. please try again.

Explanation:

Step1: Identify the solid and relevant formulas

The solid appears to be a triangular - based pyramid. The volume formula for a triangular - based pyramid is $V=\frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height. The surface area of a triangular - based pyramid is the sum of the areas of the base and the three lateral faces. First, find the area of the base. If the base is a triangle with base $b = 16$ ft and height $h_{base}=15$ ft, then the area of the base $B=\frac{1}{2}\times16\times15 = 120$ $ft^{2}$. The height of the pyramid $h = 16$ ft.

Step2: Calculate the volume

Using the volume formula $V=\frac{1}{3}Bh$, substitute $B = 120$ $ft^{2}$ and $h = 16$ ft. So, $V=\frac{1}{3}\times120\times16=640$ $ft^{3}$.

Step3: Calculate the areas of the lateral faces

There are three lateral faces. Two of the lateral - face triangles have base $b_1 = 16$ ft and height $h_1 = 17$ ft, and the third has base $b_2=12$ ft (from the right - triangle base calculation: using Pythagorean theorem, if the base of the base - triangle is 16 ft and height is 15 ft, the third side of the base - triangle is $\sqrt{16^{2}+15^{2}} = \sqrt{256 + 225}=\sqrt{481}\approx22$ ft). The area of a triangle is $A=\frac{1}{2}bh$.
The area of the first two lateral faces: $A_1=\frac{1}{2}\times16\times17 = 136$ $ft^{2}$ each. The area of the third lateral face: First, find the length of the base of the third lateral face. Using the Pythagorean theorem for the base triangle: $a = 16$, $b = 15$, then $c=\sqrt{16^{2}+15^{2}}= \sqrt{481}\approx22$ ft. Let's assume the height of the third lateral face is also 17 ft. So $A_2=\frac{1}{2}\times22\times17 = 187$ $ft^{2}$.
The total surface area $SA=B + 2A_1+A_2=120+2\times136 + 187=120 + 272+187 = 579$ $ft^{2}$.

Answer:

Volume: $640$ $ft^{3}$
Surface area: $579$ $ft^{2}$