An observatory has the shape of a right circular cylinder surmounted by a hemisphere.

Question

A cylinder and a half sphere are given. The figures are oriented so that the base of the half sphere is aligned with the top base of the cylinder. The bottom base of the cylinder has a point at its center. The distance from the bottom base of the cylinder to the top base of the cylinder is labeled 50 ft. The radius of the bottom base of the cylinder is labeled 12 ft. Consider the surface area of the observatory, if the radius of the cylinder is 12 ft and its altitude measures 50 ft.

a. Find the exact lateral area (in square feet) of the cylinder.

b. Find the exact surface area (in square feet) of the curved part of the hemisphere.

c. What is the surface area (in square feet) of the observatory? Use the calculator value of 𝜋. (Round your answer to two decimal places.)

d. If 1 gallon of paint covers 400 ft2 and you can only purchase full gallons of paint, how many gallons must be purchased to paint the surface if it requires two coats? (Round your answer up to the nearest whole number.)

Answer 1

LSA_cylinder = 2πrh

LSA - lateral surface area

r - radius of the base of the the cylinder

h - altitude

r = 12 (ft)

h = 50 (ft)

LSA_cylinder = 2π•12•50 = 3,769.91 (ft²)

LSA_hemisphere = 2πr²

LSA - lateral surface area

r - radius of the base

LSA_hemisphere = 2π•12² = 904.78 (ft²)

LSA_observatory = LSA_cylinder + LSA_hemisphere = 3,769.91 + 904.78 = 4,674.69 (ft²)

Calculating the number of gallons of paint needed to put 2 layers of paint:

P = 2 • LSA_observatory/400 = 2 • 4,674.69/400 = 23.37 (gallons)

P = 24 (gallons) ← needed to buy