Solution:

Let r₁ and r₂ be the radius of the two cylinders and h₁ and h₂ be their heights.

Given ratio of the diameter = 3:4

Then the ratio of radius r₁:r₂ = 3:4

Given volume of both jars are same.

r₁²h₁ = r₂²h₂

h₁/h₂ = r₂²/ r₁²

h₁/h₂ = 4²/3² = 16/9

Hence the ratio of the heights are 16:9.

17. A rectangular sheet of tin foil of size 30 cm × 18 cm can be rolled to form a cylinder in two ways along length and along breadth. Find the ratio of volumes of the two cylinders thus formed.

Solution:

Given size of the sheet = 30 cm×18 cm

If we roll it lengthwise, base circumference, 2r = 30

2×(22/7)r = 30

r = 30×7/2×22 = 210/44 = 105/22 cm

Height, h = 18 cm

Volume of the cylinder, V₁ = r²h

= (22/7)×(105/22)²×18

= 15×105×9/11

If we roll it breadthwise, base circumference, 2r = 18

2×(22/7)r = 18

r = 18×7/2×22 = 126/44 = 63/22 cm

Height, h = 30 cm

Volume of the cylinder, V₂ = r²h

= (22/7)×(63/22)²×30

= 9×63×15/11

V₁/V₂ = (15×105×9/11)÷( 9×63×15/11)

= (15×105×9/11)×(11/9×63×15)

= 105/63

= 15/9

= 5/3

Ratio of the volumes of two cylinders is 5:3.