Question

select the correct answer.
an agricultural silo is 30 feet across. the bottom shape is 33 feet tall, and the top shape is 8 feet tall at the peak.
image of silos
what is the visible area of one silo, to the nearest square foot?
5,325 square feet
3,911 square feet
2,356 square feet
6,032 square feet

Explanation:

Step1: Identify the shapes (cylinder + cone)

The silo has a cylindrical bottom and a conical top.

• Cylinder: diameter \( d = 30 \) ft, so radius \( r=\frac{30}{2}=15 \) ft, height \( h_{\text{cyl}} = 33 \) ft.
• Cone: radius \( r = 15 \) ft, height \( h_{\text{cone}} = 8 \) ft.

Step2: Calculate lateral (curved) surface area of cylinder

Formula: \( A_{\text{cyl}} = 2\pi rh_{\text{cyl}} \)
Substitute \( r = 15 \), \( h_{\text{cyl}} = 33 \):
\( A_{\text{cyl}} = 2\pi(15)(33) = 990\pi \approx 990 \times 3.1416 \approx 3110.18 \) sq ft.

Step3: Calculate lateral (curved) surface area of cone

First, find slant height \( l \) of cone: \( l=\sqrt{r^2 + h_{\text{cone}}^2}=\sqrt{15^2 + 8^2}=\sqrt{225 + 64}=\sqrt{289}=17 \) ft.
Formula: \( A_{\text{cone}}=\pi rl \)
Substitute \( r = 15 \), \( l = 17 \):
\( A_{\text{cone}}=\pi(15)(17)=255\pi \approx 255 \times 3.1416 \approx 801.11 \) sq ft.

Step4: Total visible area (sum of cylinder and cone lateral areas)

\( A_{\text{total}} = A_{\text{cyl}} + A_{\text{cone}} \approx 3110.18 + 801.11 \approx 3911.29 \) sq ft.

Answer:

3,911 square feet