Question

The ratio of the surface area of two spheres is 3:2. The volume of the larger sphere is 2,916 in3. What is the volume of the smaller sphere?

Answer 1

The volume of the smaller sphere is 864 in³.

Explanation:

Let
`r_1` are
`r_2` are the radius of the spheres,

Thus, the surface area of the first sphere,

`A_1=4\pi (r_1)^2`

And, the surface area of the second sphere,

`A_2=4\pi (r_2)^2`

According to the question,

`(A_1)/(A_2)=(3)/(2)`

`(4\pi (r_1)^2)/(4\pi (r_2)^2)=(3)/(2)`

`\implies (r_1)/(r_2)=(3)/(2)-------(1)`

Now,

The volume of first sphere,

`V_1=(4)/(3)\pi (r_1)^3`

And, the volume of second sphere,

`V_2=(4)/(3)\pi (r_2)^3`

`\implies (V_1)/(V_2)=((4)/(3)\pi (r_1)^3)/((4)/(3)\pi (r_2)^3)</p><p>[tex]=((r_1)/(r_2))^3`

From equation (1),

`(V_1)/(V_2)=(27)/(8)`

Given,

`V_1=2,916\text{ cube inches}`

`\implies (2,916)/(V_2)=(27)/(8)`

[tex]8\times 2,916=27V_2/tex]

[tex]23328=27V_2\implies V_2=864\text{ cube in}/tex]

Hence, the volume of the smaller sphere is 864 in³.

Answer 2

If the ratio of the radii of the two spheres is the square root of the given ratio of their surface area. That is, the ratio of the radii is sqrt 3: sqrt 2. The ratio of their volumes, on the other hand, is the cube of the ratio of their radii. That is 3^3/2 : 2^3/2. Letting x be the volume of the smaller sphere, the proportion becomes,
3^3/2 / 2^3/2 = 2,916 / x
The volume of the smaller sphere is approximately 1,587.27.