If the radius of the base of a right circular cylinder is decreased by 30% and its height is increased by 224%, then what is the percentage increase (closest integer) in its volume?

Given:

The radius of the base of a right circular cylinder is decreased by 30%.

The height is increased by 224%.

Formula used:

Volume of a cylinder = πr²h, where r is the radius and h is the height.

Calculation:

If the original radius is r and the original height is h, the original volume of the cylinder is:

Original volume = πr²h

If the radius is decreased by 30%, the new radius becomes 70% of the original radius:

New radius = 0.7r

If the height is increased by 224%, the new height becomes 324% of the original height:

New height = 3.24h (since 100% + 224% = 324%, and 324% of h is 3.24h).

New volume = π × (0.7r)² × 3.24h = π × 0.49r² × 3.24h

New volume = 1.5916 × πr²h

The percentage increase in volume is given by:

Percentage increase = [(New volume - Original volume) / Original volume] × 100

⇒ Percentage increase = [(1.5916 × πr²h - πr²h) / πr²h] × 100

⇒ Percentage increase = (0.5916 / 1) × 100 = 59.16%

∴ The percentage increase in volume is approximately 59% (rounded to the nearest integer).