Cylinder-and-Hemisphere Silo Capacity and Paint Order
A farmer is replacing a grain silo made of a right circular cylinder with a hemispherical roof. The cylinder has radius 6.0 ft and height 18.0 ft; the hemisphere sits on top with the same radius. The silo sits on a concrete floor, so the bottom circle is not painted. The sales sheet lists the capacity as 3,000 cubic feet and says the exterior curved surface can be painted with 3 gallons if one gallon covers 350 square feet. The farmer also wants a tabletop model in which every dimension is 1.5 times the corresponding dimension of the real silo, and has been told that the model will hold 1.5 times as much and need 1.5 times as much paint. The farmer needs to know whether the capacity claim, paint order, and scaling statements are correct before signing the order. Use π ≈ 3.1416.
1. Compute the silo's volume and decide whether the 3,000-cubic-foot capacity claim is accurate.
Read the worked answer
The solid is a cylinder plus a hemisphere, both with radius 6 ft.
Cylinder volume: V_cyl = πr²h = π(6²)(18) = 648π ≈ 648(3.1416) = 2035.76 ft³.
Hemisphere volume: V_hemi = (2/3)πr³ = (2/3)π(6³) = (2/3)π(216) = 144π ≈ 452.39 ft³.
Total volume: V ≈ 2035.76 + 452.39 = 2488.15 ft³.
The 3,000-ft³ claim overstates the capacity by about 511.85 ft³. The actual capacity is about 82.9% of the claimed value, so the sales sheet should not be used as if it were exact.
2. Compute the exterior curved area that must be painted and decide whether 3 gallons is enough for one coat.
Read the worked answer
Only the curved cylindrical side and the curved hemispherical roof are painted; the floor circle is excluded.
Cylinder lateral area: A_cyl = 2πrh = 2π(6)(18) = 216π ≈ 678.58 ft².
Hemisphere curved area: A_hemi = 2πr² = 2π(6²) = 72π ≈ 226.19 ft².
Total painted area: A ≈ 678.58 + 226.19 = 904.78 ft².
At 350 ft² per gallon, the number of gallons is 904.78/350 ≈ 2.59 gallons.
So 3 gallons is enough for one coat, provided the surface is not unusually porous and no large waste occurs. Three gallons cover 3(350) = 1050 ft², leaving some margin. If two coats are required, 3 gallons is not enough; about 5.18 gallons, or 6 gallons, would be needed.
3. Evaluate the scaling statements for the 1.5-times tabletop model: volume 1.5 times and paint 1.5 times. Give the correct factors and the model's approximate volume and paint area.
Read the worked answer
If every linear dimension is multiplied by 1.5, areas are multiplied by 1.5² = 2.25 and volumes by 1.5³ = 3.375. The statements are therefore both wrong.
Model volume ≈ 2488.15(3.375) ≈ 8397.50 ft³ in the same units. Model painted area ≈ 904.78(2.25) ≈ 2035.76 ft².
At 350 ft² per gallon, the model would need about 2035.76/350 ≈ 5.82 gallons, so 6 gallons for one coat. The model does not need 1.5 times the paint; it needs 2.25 times the paint area.