Question

5. If the volumes of two cones are in the ratio of 1:4 and their diameters are in the ratio of 4:5, find the ratio of their heights.
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Answer 1

Answer:

Ratio of their heights = 25:64.

Step-by-step explanation:

Given:

• The volumes of two cones are in the ratio of 1:4.
• Their diameters are in the ratio of 4:5.

To find:

• Ratio of their heights.

Solution:

Let the height of the 1st cone be h unit and the height of the 2nd cone be h' unit.

Let ,

• Diameter of 1st cone = 4x unit
• Diameter of 2nd cone = 5x unit

Then,

• Radius of 1st cone = 4x/2 unit = 2x unit
• Radius of 2nd cone = 5x/2 unit

We know that,

${\boxed{\sf{Volume\:of\:cone=\dfrac{1}{3}\pi\:r^2h}}}$

Volume of 1st cone ,

= ⅓ πr²h

= ⅓ π(2x)²h unit ³

= ⅓ × 4x²πh unit ³

&

Volume of 2nd cone,

= ⅓ πr²h'

= ⅓ π × (5x/2)² × h' unit ³

= ⅓ × (25x²/4)πh' unit ³

According to the question,

⅓ × 4x²πh : ⅓ × (25x²/4)πh' = 1:4

→ ⅓ × 4x²πh / ⅓ × (25x²/4)πh' = 1/4

→ 16h/25h' = 1/4

→ h/h' = 25 /(16×4)

→ h/h' = 25/64

→ h : h' = 25:64

Therefore, the ratio of their heights is 25:64.