An observatory has the shape of a right circular cylinder topped by a hemisphere. The radius of the cylinder is 8 ft and its altitude measures 26 ft
What is the approximate surface area of the observatory? Round to the nearest foot. Show all work to receive full credit. (Hint: Remember the top and bottom of the cylinder will not be painted, so do not include them in your surface area. However, note that the hemispherical dome will be painted.)
B. If 1 gallon of paint covers 300 ft2, how many gallons are needed to paint the surface if it requires three coats? Round up to the nearest gallon. Show all work to receive full credit.
To find the surface area, we need to calculate the lateral surface area of the cylinder and the surface area of the hemisphere.
1. Lateral surface area of the cylinder:
The formula for the lateral surface area of a cylinder is given by LSA = 2πrh, where π is approximately equal to 3.14, r is the radius, and h is the altitude.
Plugging in the values, we get LSA = 2 * 3.14 * 8 * 26 = 1312.32 ft².
2. Surface area of the hemisphere:
The formula for the surface area of a hemisphere is given by SA = 2πr², where r is the radius.
Plugging in the value, we get SA = 2 * 3.14 * 8² = 401.92 ft².
3. Total surface area:
The total surface area of the observatory is the sum of the lateral surface area of the cylinder and the surface area of the hemisphere.
Total surface area = LSA + SA = 1312.32 + 401.92 = 1714.24 ft².
Therefore, the approximate surface area of the observatory is 1714 ft² (rounded to the nearest foot).
B. To find the number of gallons needed to paint the surface, we need to divide the total surface area by the coverage of one gallon of paint.
1. Coverage per gallon of paint: 300 ft²
2. Total surface area: 1714 ft²
3. Gallons needed for one coat: Total surface area ÷ Coverage per gallon = 1714 ft² ÷ 300 ft²/gallon ≈ 5.71 gallons
Since we need three coats of paint, we need to multiply the gallons needed for one coat by 3.
Gallons needed for three coats ≈ 5.71 gallons × 3 = 17.13 gallons.
Therefore, approximately 18 gallons (rounded up to the nearest gallon) of paint are needed to paint the surface with three coats.
To find the surface area of the observatory, we need to calculate the surface area of the cylindrical part and the hemispherical dome separately.
Surface area of the cylinder = 2πrh
where r is the radius of the cylinder and h is the altitude of the cylinder
Given: r = 8 ft and h = 26 ft
Substituting the values, we get:
Surface area of the cylinder = 2π(8 ft)(26 ft)
= 416π ft^2
Surface area of the hemisphere = 2πr^2
where r is the radius of the hemisphere
Given: r = 8 ft
Substituting the value, we get:
Surface area of the hemisphere = 2π(8 ft)^2
= 128π ft^2
Since the top and bottom of the cylinder are not painted, we only need to consider the surface area of the cylindrical part of the observatory and the hemispherical dome.
Total surface area = Surface area of the cylinder + Surface area of the hemisphere
= 416π ft^2 + 128π ft^2
≈ 544π ft^2
To round to the nearest foot, we need to find the approximate value of the surface area in terms of feet. 1π ft^2 is approximately equal to 3.14 ft^2. So, the approximate surface area in feet is:
544π ft^2 ≈ 544(3.14) ft^2
≈ 1708.16 ft^2
≈ 1708 ft^2 (rounded to the nearest foot)
To find the number of gallons of paint needed, we need to divide the surface area of the observatory by the coverage of 1 gallon of paint.
Given: 1 gallon of paint covers 300 ft^2
Number of gallons needed = Total surface area / Coverage of 1 gallon of paint
= 1708 ft^2 / 300 ft^2
≈ 5.6933 gallons
Since we need to apply three coats of paint, we need to multiply the number of gallons by 3.
Number of gallons needed for three coats = 5.6933 gallons × 3
= 17.08 gallons
Rounding up to the nearest gallon, we get:
Number of gallons needed for three coats ≈ 18 gallons
As = 2(pi*r^2) + pi*D*h.
As = 2(3.14*64) + 3.14*16*26,
As = 401.92 + 1306.24 = 1708ft^2.