If the radius of the base of a right circular cylinder is decreased by 24% and its height is increased by 262%, then what is the percentage increase (closest integer) in its volume?

Given:

Initial radius (r₁) = r

Initial height (h₁) = h

New radius (r₂) = r × (1 - 0.24) = 0.76r

New height (h₂) = h × (1 + 2.62) = 3.62h

Formula used:

Volume of a cylinder = πr²h

Percentage change in volume = \(\dfrac{\text{New Volume - Old Volume}}{\text{Old Volume}} \times 100\)

Calculations:

Initial volume (V₁) = πr²h

New volume (V₂) = π(0.76r)²(3.62h)

⇒ V₂ = π(0.5776r²)(3.62h)

⇒ V₂ = π × 2.090912r²h

Percentage change in volume:

⇒ \(\dfrac{V_2 - V_1}{V_1} \times 100\)

⇒ \(\dfrac{(2.090912πr^2h) - (πr^2h)}{πr^2h} \times 100\)

⇒ \(\dfrac{2.090912 - 1}{1} \times 100\)

⇒ 1.090912 × 100

⇒ 109.09%

∴ The percentage increase in volume is approximately 109%.

∴ The correct answer is option (4).