Question

The ratio of volumes of two cones is 4:5 and the ratio of their radii is 2:3. Find the ratio of their heights.
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Answer 1

The ratio of the heights = 9:5

Step by step explanation :-

Given :-

■ The ratio of the volumes of two cones is 4:5

■ The ratio of their radii = 2:3

To find :-

■The ratio of their heights

Solution :-

Given that

The ratio of the volumes of two cones

= 4:5

Let they be 4X cubic units and 5X cubic units

The volume of the first cone

= 4X cubic units

The volume of the second cone

= 5X cubic units

The ratio of their radii = 2:3

Let they be 2Y units and 3Y units

The radius of the first cone = 2Y units

The radius of the second cone = 3Y units

Let the height of the first cone be h units

Let the height of the second cone be

H units

We know that

■ Volume of a cone = (1/3)πr²h cubic units

Volume of the first cone = 4X cubic units

=> (1/3)π(2Y)²×h = 4X

=> (1/3)π(4Y²)h = 4X

=> 4πY²h/3 = 4X

=> 4πY²h = 4X×3

=> 4πY²h = 12X

=> h = 12X/4πY²

=> h = 3X/πY²

The height of the first cone = 3X/(πY²) units

The volume of the second cone = 5X units

=> (1/3)π(3Y)²×H = 5X

=> (1/3)π(9Y²)H = 5X

=> 9πY²H/3 = 5X

=> 3πY²H = 5X

=> H = 5X/3πY²

The height of the first cone = 5X/(3πY²) units

Now,

The ratio of the heights of the two cones = h : H

=> 3X/(πY²) : 5X/(3πY²)

=> [3X/(πY²)] /[ 5X/(3πY²)]

=> 3X/(5X/3)

=> (3X×3)/5X

=> 9X/5X

=> 9/5

=> 9:5

The required ratio = 9:5

Answer :-

♦ The ratio of the heights of the two cones is 9:5

Used formulae:-

• Volume of a cone = (1/3)πr²h cubic units

• r = radius
• h = height
• π = 22/7