Question

Find the ratio of volumes of two right circular cylinders if the ratio of
their heights is 1.2 and ratio of their circumference of bases is 3:4.​

Answer 1

Answer: 9:32

Solution:

Let the height of the two cylinders = h₁ and h₂ respectively.

Let the radii of the two cylinders = r₁ and r₂ respectively

According to the question,

$\frac{2\pi r_1}{2\pi r_2} = \frac{3}{4}$

⇒ $\frac{r_1}{r_2} = \frac{3}{4}$    -------------------------------------------(1)

And

$\frac{h_1}{h_2} = \frac{1}{2}$     ---------------------------------------------(2)

Let the volume of first and second cylinder = V₁ and V₂ respectively.

V₁ = πr₁²h₁

V₂ = πr₂². h₂

⇒ $\frac{V_{1} }{V_2} = \frac{ \pi r_1^{2} h_1} { \pi r_2^{2} .h_2}$

⇒ $\frac{V_1}{V_2} =( \frac{r_1}{r_2} )^{2} X (\frac{h_1}{h_2} )$

⇒ $\frac{V_1}{V_2} = (\frac{3}{4} )^{2} X (\frac{1}{2})$    -----------------------------------------[By (1) and (2)]

⇒ $\frac{V_1}{V_2} = \frac{9}{16} X \frac{1}{2}$

⇒ $\frac{V_1}{V_2} = \frac{9}{32}$

Hence, $\frac{V_1}{V_2} = \frac{9}{32}$

⇒ V₁ : V₂ = 9 : 32

The ratio of the volumes of the two given cylinders = V₁ : V₂ = 9 : 32

Hope you understood.

Thank You.