If base diameter of a cylinder is increased by 50%, then by how much per cent its height must be decreased so as to keep its volume unaltered?

Let radius, $$r = 10$$ and height, $$h = 10$$

If diameter is increased by 50%, then radius is also increased by 50%, since they both are directly proportional.

Original Volume of cylinder = $$V = \pi r^2 h$$

=> $$V = \pi (10)^2 (10) = 1000 \pi$$

New radius, $$r' = 10 + \frac{50}{100} \times 10 = 15$$

Let new height = $$h'$$

Since, volume is unaltered, => New volume = $$V' = \pi (r')^2 h' = 1000 \pi$$

=> $$(15)^2 h' = 1000$$

=> $$h' = \frac{1000}{225} = \frac{40}{9}$$

$$\therefore$$ % Change in height = $$\frac{10 - \frac{40}{9}}{10} \times 100$$

= $$\frac{50}{9} \times 10 = 55.56 \%$$