Question

problem 7
a silo is a large cylindrical container used on farms to hold grain.
on estaban’s farm, a silo has a cone - shaped spout on the bottom to regulate the flow of grain going out.
the diameter of the silo is 8 feet. the cylindrical part of the silo has a height of 12 feet, and the height of the entire silo is 16 feet.
approximately how many cubic feet of grain can the entire silo hold?

Explanation:

Step1: Find the height of the cone

The total height of the silo is 16 ft, and the height of the cylinder is 12 ft. So the height of the cone \( h_{cone} = 16 - 12 = 4 \) ft. The radius of the silo (since diameter is 8 ft) \( r = \frac{8}{2} = 4 \) ft.

Step2: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \). Substituting \( r = 4 \) ft and \( h = 12 \) ft, we get \( V_{cylinder} = \pi \times 4^2 \times 12 = \pi \times 16 \times 12 = 192\pi \) cubic feet.

Step3: Calculate the volume of the cone

The formula for the volume of a cone is \( V_{cone} = \frac{1}{3}\pi r^2 h \). Substituting \( r = 4 \) ft and \( h = 4 \) ft, we get \( V_{cone} = \frac{1}{3}\pi \times 4^2 \times 4 = \frac{1}{3}\pi \times 16 \times 4 = \frac{64}{3}\pi \) cubic feet.

Step4: Calculate the total volume of the silo

The total volume \( V = V_{cylinder} + V_{cone} = 192\pi + \frac{64}{3}\pi = \frac{576\pi + 64\pi}{3} = \frac{640\pi}{3} \). Approximating \( \pi \approx 3.14 \), we get \( V \approx \frac{640 \times 3.14}{3} \approx \frac{2009.6}{3} \approx 669.87 \) cubic feet. Rounding to a reasonable approximation, it's about 670 cubic feet (or more accurately, if we calculate \( \frac{640\times3.1416}{3}\approx 670.21 \)).

Answer:

Approximately \(\boxed{670}\) cubic feet (or more precisely around 670.21, which can be rounded as needed)