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Difficulty: MediumThree-Dimensional Geometry: Volume and Surface Area

A right circular cylinder has a base radius of rr and a height of hh. If the base radius is doubled and the height is halved to create a new right circular cylinder, which of the following statements must be true? Select all that apply.

  1. The volume of the new cylinder is twice the volume of the original cylinder.Answer
  2. The lateral surface area of the new cylinder is equal to the lateral surface area of the original cylinder.Answer
  3. C
    The total surface area of the new cylinder is twice the total surface area of the original cylinder.
  4. D
    The volume of the new cylinder is four times the volume of the original cylinder.
  5. The area of the circular base of the new cylinder is four times the area of the circular base of the original cylinder.Answer

Answer

The correct statements are: the volume of the new cylinder is twice the volume of the original cylinder, the lateral surface area of the new cylinder is equal to the lateral surface area of the original cylinder, and the area of the circular base of the new cylinder is four times the area of the circular base of the original cylinder.
The volume formula V=πr2hV = \pi r^2 h scales by 22×12=22^2 \times \frac{1}{2} = 2, so volume doubles. The lateral surface area formula L=2πrhL = 2\pi r h scales by 2×12=12 \times \frac{1}{2} = 1, remaining unchanged. The base area formula A=πr2A = \pi r^2 scales by 22=42^2 = 4, quadrupling the area.

Step-by-Step Solution

1
Write down the original formulas for a cylinder with radius rr and height hh.
Base Area A1=πr2A_1 = \pi r^2, Lateral Surface Area L1=2πrhL_1 = 2\pi r h, Total Surface Area T1=2πrh+2πr2T_1 = 2\pi r h + 2\pi r^2, Volume V1=πr2hV_1 = \pi r^2 h.
Establish baseline geometric properties.
2
Substitute the new dimensions r′=2rr' = 2r and h′=h2h' = \frac{h}{2} into each formula.
New Base Area A2=π(2r)2=4πr2=4A1A_2 = \pi (2r)^2 = 4\pi r^2 = 4A_1.
New Lateral Surface Area L2=2π(2r)(h2)=2πrh=L1L_2 = 2\pi (2r) \left(\frac{h}{2}\right) = 2\pi r h = L_1.
New Volume V2=π(2r)2(h2)=π(4r2)(h2)=2πr2h=2V1V_2 = \pi (2r)^2 \left(\frac{h}{2}\right) = \pi (4r^2) \left(\frac{h}{2}\right) = 2\pi r^2 h = 2V_1.
Calculate the updated values for each geometric property.
3
Evaluate each statement against the calculated values.
Volume doubles (V2=2V1V_2 = 2V_1), lateral surface area remains equal (L2=L1L_2 = L_1), and base area quadruples (A2=4A1A_2 = 4A_1).
Determine which options state correct mathematical relationships.

Key Concept

Scaling effects on 3D geometry formulas for right circular cylinders
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Topics

Three-Dimensional Geometry: Volume and Surface Area

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