Two right circular solid cylinders have radii in the ratio 3 : 5 and heights in the ratio 2 : 3. Find the ratio between their :

(i) curved surface areas.

(ii) volumes.

(i) According to question,

r₁ : r₂ = 3 : 5

Let, r₁ = 3x and r₂ = 5x.

h₁ : h₂ = 2 : 3

Let, h₁ = 2y and h₂ = 3y.

$\dfrac{\text{CSA of 1st cylinder}}{\text{CSA of 2nd cylinder}} = \dfrac{2πr$1h1}{2πr2h2} \\[1em] = \dfrac{r1h1}{r2h2} \\[1em] = \dfrac{3x \times 2y}{5x \times 3y} \\[1em] = \dfrac{6xy}{15xy} \\[1em] = \dfrac{2}{5} \\[1em] = 2 : 5.

Hence, ratio between curved surface area = 2 : 5.

(ii)

$\dfrac{\text{Vol. of 1st cylinder}}{\text{Vol. of 2nd cylinder}} = \dfrac{πr$1^2h1}{πr2^2h2} \\[1em] = \dfrac{r1^2h1}{r2^2h2} \\[1em] = \dfrac{(3x)^2 \times 2y}{(5x)^2 \times 3y} \\[1em] = \dfrac{18x^2y}{75x^2y} \\[1em] = 6 : 25.

Hence, ratio between volume = 6 : 25.