Question

the ratio of radii of two spheres is 2:3 find the ratio of their surface areas and volumes

Answer 1

The ratio of surface area of the sphere is 4 : 9 and the ratio of volume of the sphere is 8 : 27

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Answer 2

Given : The ratio of radii of two spheres is 2:3

To find : The ratio of their surface areas and volumes.

Solution :

We can simply solve this mathematical problem by using the following mathematical process. (our goal is to calculate the ratio of their surface areas and the ratio of their volumes)

Let, their radii = 2x and 3x

(obtained from the ratio 2:3)

Now, surface area of a sphere = 4 × π × (radius)²

So,

• Surface area of the first sphere = 4 × π × (2x)² = (4π × 4x²)
• Surface area of the second sphere = 4 × π × (3x)² = (4π × 9x²)

And, the ratio of their surface areas :

= (4π × 4x²) : (4π × 9x²)

= 4x² : 9x²

= 4:9

Now, volume of a sphere = ⁴/₃ × π × (radius)³

So,

• Volume of the first sphere = ⁴/₃ × π × (2x)³ = (⁴/₃ × π × 8x³)
• Volume of the second sphere = ⁴/₃ × π × (3x)³ = (⁴/₃ × π × 27x³)

And, the ratio of their volumes :

= (⁴/₃ × π × 8x³) : (⁴/₃ × π × 27x³)

= 8x³ : 27x³

= 8:27

Hence, the ratio of their surface areas is 4:9 and the ratio their volumes is 8:27