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The radius of a cylinder is 3.5 ft. The height is 14 ft. Find the surface area and volume of the cylinder to the nearest tenth of a foot. Show your work.
To find the surface area of a cylinder, we need to calculate the area of the two circular bases and the area of the curved surface.
Step 1: Calculate the area of the circular bases.
The formula for the area of a circle is A = πr^2, where r is the radius.
Given that the radius is 3.5 ft, we can use the formula to find the area of one circular base:
A_base = π(3.5)^2
A_base = π(12.25) (rounded to two decimal places)
A_base ≈ 38.48 ft^2
Since there are two circular bases, we multiply this area by 2 to get the total area of the bases:
Total base area = 2(38.48)
Total base area ≈ 76.96 ft^2
Step 2: Calculate the area of the curved surface.
The formula for the curved surface area of a cylinder is A_curved = 2πrh, where r is the radius and h is the height.
Given that the radius is 3.5 ft and the height is 14 ft, we can use the formula to find the curved surface area:
A_curved = 2π(3.5)(14)
A_curved ≈ 308.88 ft^2
Step 3: Add the areas of the bases and the curved surface to find the total surface area.
Total Surface Area = Total base area + Curved surface area
Total Surface Area ≈ 76.96 + 308.88
Total Surface Area ≈ 385.84 ft^2
Therefore, the surface area of the cylinder is approximately 385.84 ft^2.
Now, let's find the volume of the cylinder.
Step 4: Calculate the volume of the cylinder.
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Given that the radius is 3.5 ft and the height is 14 ft, we can use the formula to find the volume:
V = π(3.5)^2(14)
V = π(12.25)(14) (rounded to two decimal places)
V ≈ 615.75 ft^3
Therefore, the volume of the cylinder is approximately 615.75 ft^3.
Surface Area:
Surface Area = 2πr2 + 2πrh
= 2π(3.5 ft)2 + 2π(3.5 ft)(14 ft)
= 2π(12.25 ft2) + 2π(49 ft)
= 24.5π ft2 + 98π ft2
= 122.5π ft2
= 384.7 ft2 (to the nearest tenth of a foot)
Volume:
Volume = πr2h
= π(3.5 ft)2(14 ft)
= π(12.25 ft2)(14 ft)
= 171.5π ft3
= 545.2 ft3 (to the nearest tenth of a foot)