A right circular cylinder has base radius r and height h. The height of the cylinder is in… — GRE General Test

A right circular cylinder has base radius $r$ and height $h$ . The height of the cylinder is increased by $50\%$ , and its base radius is decreased by $20\%$ . Which of the following statements about the modified cylinder compared to the original cylinder must be true? Select all that apply.

The volume of the cylinder decreases by $4\%$ .Answer
The lateral surface area of the cylinder increases by $20\%$ .Answer
C
The combined area of the two circular bases decreases by $20\%$ .
If $h = r$ , the total surface area of the cylinder decreases.Answer
E
If $h = 4r$ , the total surface area of the cylinder decreases.

Answer

The statements confirming a

4\%

volume decrease, a

20\%

lateral surface area increase, and a total surface area decrease when

h = r

are all true.

The volume of the modified cylinder decreases by

4\%

because

0.8^2 \times 1.5 = 0.96

. The lateral surface area increases by

20\%

because

0.8 \times 1.5 = 1.20

. When

h = r

, the original total surface area

4\pi r^2

reduces to

3.68\pi r^2

, confirming a decrease.

Step-by-Step Solution

Express original and modified dimensions algebraically.

Original: radius

= r

, height

= h

. Modified: radius

= 0.8r

, height

= 1.5h

Decreasing radius by

20\%

multiplies it by

1 - 0.20 = 0.80

, while increasing height by

50\%

multiplies it by

1 + 0.50 = 1.50

Calculate the ratio of modified volume to original volume.

V_{new} = \pi (0.8r)^2 (1.5h) = 0.64 \times 1.5 \pi r^2 h = 0.96 V_{old}

, indicating a

4\%

volume decrease.

Volume of a cylinder is given by

V = \pi r^2 h

Calculate the ratio of modified lateral surface area to original lateral surface area.

L_{new} = 2\pi (0.8r)(1.5h) = 2.4\pi r h = 1.20 L_{old}

, indicating a

20\%

lateral surface area increase.

Lateral surface area of a cylinder is given by

L = 2\pi r h

Evaluate the base area change and total surface area under specific height-to-radius ratios.

Base area becomes

(0.8)^2 = 0.64

of original (

36\%

decrease). For

h = r

, total surface area changes from

4\pi r^2

3.68\pi r^2

(decrease). For

h = 4r

, total surface area changes from

10\pi r^2

10.88\pi r^2

(increase).

Total surface area is the sum of lateral surface area and base area,

A = 2\pi r^2 + 2\pi r h

Key Concept

Impact of dimensional scaling on volume, lateral surface area, and total surface area of cylinders

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