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Difficulty: EasyThree-Dimensional Geometry: Volume and Surface Area

A right circular cylinder has a base radius of 33 units and a height of 44 units. Which of the following statements regarding this cylinder are true? Select all such statements.

  1. The volume of the cylinder is 36π36\pi cubic units.Answer
  2. The total surface area of the cylinder is 42π42\pi square units.Answer
  3. C
    The volume of the cylinder is 24π24\pi cubic units.
  4. D
    The total surface area of the cylinder is 36π36\pi square units.
  5. E
    The ratio of the area of one circular base to the lateral surface area is 11 to 22.

Answer

The correct statements are that the volume of the cylinder is 36π36\pi cubic units and the total surface area of the cylinder is 42π42\pi square units.
The volume of a cylinder with radius 33 and height 44 is V=πr2h=π(3)2(4)=36πV = \pi r^2 h = \pi(3)^2(4) = 36\pi. The total surface area is A=2πrh+2πr2=2π(3)(4)+2π(3)2=24π+18π=42πA = 2\pi r h + 2\pi r^2 = 2\pi(3)(4) + 2\pi(3)^2 = 24\pi + 18\pi = 42\pi. Therefore, both statements asserting these exact values are correct.

Step-by-Step Solution

1
Calculate the volume of the right circular cylinder
Volume V=πr2h=π(32)(4)=36πV = \pi r^2 h = \pi (3^2)(4) = 36\pi
The formula for the volume of a right circular cylinder is V=πr2hV = \pi r^2 h.
2
Calculate the lateral surface area and total surface area of the cylinder
Lateral Area = 2πrh=24π2\pi r h = 24\pi; Total Area = 24π+2(π⋅32)=42π24\pi + 2(\pi \cdot 3^2) = 42\pi
Total surface area is the sum of the lateral surface area (2πrh2\pi r h) and the areas of the two circular bases (2πr22\pi r^2).
3
Evaluate the base-to-lateral surface area ratio
Ratio = 9π24π=38\frac{9\pi}{24\pi} = \frac{3}{8}
Comparing the base area (9π9\pi) to the lateral area (24π24\pi) simplifies to 3:83:8.

Key Concept

Volume and surface area formulas for right circular cylinders
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Topics

Three-Dimensional Geometry: Volume and Surface Area

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