Silo Paint Order and Volume Check
A city parks department is designing a cylindrical storage silo with a hemispherical roof. The cylindrical wall has radius 6 ft and height 14 ft, and the hemispherical roof sits exactly on top. The exterior curved wall and roof must be painted, but the flat concrete floor at the bottom is not painted. One gallon of paint covers 350 ft² and costs $38; the supplier also sells only 5-gallon buckets for $170. The department wants the least expensive way to buy enough paint for one coat, and it also needs the storage volume for a grant application. A staff member calculates the roof area as 4πr² and the volume as πr²h + (4/3)πr³, then doubles the paint order because “the roof is a full sphere cut in half.” The project manager asks for corrected surface-area and volume calculations, a purchase recommendation, and an explanation of how the paint quantity would change if the radius were doubled while the cylindrical height stayed the same.
1. Compute the exterior area to be painted and the number of gallons needed for one coat. Round up to a whole gallon.
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The painted exterior consists of the cylindrical lateral area and the curved hemispherical roof; the floor is excluded. Cylinder lateral area: 2πrh = 2π(6)(14) = 168π ≈ 527.8 ft². Hemisphere area: 2πr² = 2π(36) = 72π ≈ 226.2 ft². Total area = 168π + 72π = 240π ≈ 754.0 ft². Gallons needed = 754.0/350 ≈ 2.15, so 3 whole gallons are required for one coat.
2. Compute the silo’s volume and evaluate the staff member’s formulas. State the correct volume in exact and approximate form.
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Cylinder volume: πr²h = π(36)(14) = 504π ≈ 1583.4 ft³. Hemisphere volume: (1/2)(4/3)πr³ = (2/3)π(216) = 144π ≈ 452.4 ft³. Total volume = 504π + 144π = 648π ≈ 2035.8 ft³. The staff member used the full-sphere formulas 4πr² and (4/3)πr³ instead of the hemisphere formulas. A hemisphere has half the surface area and half the volume of a sphere, so the roof area should be 2πr² and the roof volume should be (2/3)πr³. Doubling the paint order is not justified by the geometry.
3. Recommend the cheaper paint purchase and explain how doubling the radius, with the height unchanged, changes the painted area and the volume.
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Three single gallons cost 3 × $38 = $114. The 5-gallon bucket costs $170, so buying 3 single gallons is cheaper. If the radius doubles to 12 ft while h stays 14 ft, the new lateral area is 2π(12)(14) = 336π, and the new hemisphere area is 2π(144) = 288π. The new total is 624π ≈ 1960.4 ft², compared with 240π ≈ 754.0 ft². The area ratio is 624/240 = 2.6, not 4, because only the radius doubled and the cylinder height did not. Paint needed would be 1960.4/350 ≈ 5.6, so 6 gallons. The new volume is π(144)(14) + (2/3)π(1728) = 2016π + 1152π = 3168π ≈ 9953.6 ft³. The volume ratio is 3168/648 ≈ 4.89, not 8, again because the height was unchanged.