If the heights of two cylinders are in the ratio 4:3 and their radii are in the ratio 2:3

If radii of two cylinders are in the ratio 4 : 3 and their heights are in the ratio 5: 6, find the ratio of their curved surfaces.

The ratio in radii of two cylinders = 4 : 3
and ratio in their heights = 5: 6

Let r1 and r2 be the radii and h1,h2 be their heights respectively.

∴ r1 : r2 = 4:3 and h1 : h2 = 5:6

∴ `r_1 = 4/3  "and"  (h_1)/(h_2) = 5/6`

∴ Surface area of the first cylinder = `2pir_1h_1`

and area of second cylinder = `2pir_2h_2`

`(2pir_1h_1)/(2pir_2h_2) = r_1/r_2 xx h_1/h_2 = 4/3 xx 5/6 = 20/18`

= `10/9 = 10 : 9`

∴ Ratio in their surface areas = 10 : 9

Concept: Surface Area of a Cuboid

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The radii of two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3. Calculate the ratio of their volumes.

Solution

Ratio of radii of two cylinders $=2:3$

and the ratio of heights $=5:3$

Let $r_1,h_1$ and $r_2,h_2$ be the radii and heights of two cylinders respectively.

$\therefore \frac{r_1}{r_2}=\frac{2}{3}$ and $\frac{h_1}{h_2}=\frac{5}{3}$

Now $\frac{\text{Volume~of~first~cylinder}}{\text{Volume~ of~second~cylinder}}=\frac{\pi (r_1{)}^2{h}_1}{\pi (r_2{)}^2{h}_2}$

$={\left(\frac{r_1}{r_2}\right)}^2\times \left(\frac{h_1}{h_2}\right)$

$={\left(\frac{2}{3}\right)}^2\times \frac{5}{3}$

$=\frac{4}{9}\times \frac{5}{3}=\frac{20}{27}$

Mathematics

RD Sharma

Standard VIII