The radius of a solid right circular cylinder decreases by 20% and its height increases by 10%. Find the percentage change in its:

(i) volume (ii) curved surface area

Let the original dimensions of the solid right circular cylinder be

radius = r cm and height = h cm.

Volume = πr²h.

Curved surface area = 2πrh

Now, after the changes the new dimensions are:

Radius (r') = r - $\dfrac{20}{100} \times r$ = r - 0.2r = 0.8r

Height (h') = h + $\dfrac{10}{100} \times h$ = h + 0.1h = 1.1h

So,

New volume = πr'²h'

= π(0.8r)²(1.1h)

= 0.704 πr²h.

New curved surface area = 2πr'h' = 2π(0.8r)(1.1h)

= 1.76πrh

(i) Decrease in volume = Original volume - New volume

= πr²h - 0.704 πr²h

= 0.296 πr²h

Percentage change in its volume = $\dfrac{\text{Decrease in volume}}{\text{Original volume}}$ x 100 %

= $\dfrac{0.296πr^2h}{πr^2h}$ x 100 %

= 0.296 x 100 % = 29.6 %.

Hence, decrease in volume = 29.6 %.

(ii) Decrease in curved surface area = Original curved surface area - New curved surface area

= 2πrh - 1.76πrh

= 0.24πrh.

Percentage change in its curved surface area = $\dfrac{\text{Decreased CSA}}{\text{Original CSA}}$ x 100 %

= $\dfrac{0.24πrh}{2πrh}$ x 100 %

= 0.12 x 100 %

= 12 %.

Hence, decrease in volume = 12 %.