Question

find the volume of the silo if the radius of the hemispherical top is 20 feet. use 3.14 for π.
image of silo: hemisphere on cylinder, cylinder height labeled 5 ft (possibly a typo, likely 50 ft)
the volume of the silo, rounded to the nearest thousand, is about ____ cubic feet.
the solution is blank

Explanation:

Step1: Identify the components of the silo

The silo consists of a cylinder and a hemisphere. The radius \( r \) of both the cylinder and the hemisphere is 20 feet. The height \( h \) of the cylinder is 50 feet (assuming the "50 ft" label in the diagram, as it's a common silo dimension with radius 20 and height 50).

Step2: Volume of the cylinder

The formula for the volume of a cylinder is \( V_{\text{cylinder}} = \pi r^2 h \). Substituting \( r = 20 \), \( h = 50 \), and \( \pi = 3.14 \):
\[
V_{\text{cylinder}} = 3.14 \times 20^2 \times 50
\]
\[
V_{\text{cylinder}} = 3.14 \times 400 \times 50
\]
\[
V_{\text{cylinder}} = 3.14 \times 20000 = 62800
\]

Step3: Volume of the hemisphere

The formula for the volume of a sphere is \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \), so the volume of a hemisphere is half of that: \( V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \). Substituting \( r = 20 \) and \( \pi = 3.14 \):
\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 20^3
\]
\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 8000
\]
\[
V_{\text{hemisphere}} = \frac{2 \times 3.14 \times 8000}{3}
\]
\[
V_{\text{hemisphere}} = \frac{50240}{3} \approx 16746.67
\]

Step4: Total volume of the silo

Add the volume of the cylinder and the volume of the hemisphere:
\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}}
\]
\[
V_{\text{total}} = 62800 + 16746.67 \approx 79546.67
\]

Step5: Round to the nearest thousand

Rounding 79546.67 to the nearest thousand: look at the hundreds digit (5), so we round up. 79546.67 ≈ 80000.

Answer:

80000