Question

The radii of two spheres are 4ft and 8ft. Find the volume of a sphere whose surface area is equal to the sum of the surface areas of these two spheres.

Answer 1

To find the volume of a sphere whose surface area is equal to the sum of the surface areas of two spheres, calculate the surface areas of the two spheres using the formula A = 4(pi)(r)^2. Then, set up the equation (4/3)(pi)(r)^3 = sum of surface areas and solve for r. Finally, use the formula V = (4/3)(pi)(r)^3 to find the volume of the sphere.

Step-by-step explanation:

To find the volume of a sphere whose surface area is equal to the sum of the surface areas of two spheres, we need to first find the surface areas of the two spheres. The formula for the surface area of a sphere is given by A = 4(pi)(r)^2. Given that the radii of the two spheres are 4ft and 8ft, we can calculate their surface areas as follows:

A1 = 4(pi)(4)^2 = 64(pi) ft^2

A2 = 4(pi)(8)^2 = 256(pi) ft^2

The sum of the two surface areas is A1 + A2 = 64(pi) + 256(pi) = 320(pi) ft^2.

To find the volume of the sphere whose surface area is equal to the sum of the surface areas of the two spheres, we use the formula for the volume of a sphere: V = (4/3)(pi)(r)^3. Since the surface area is equal to 320(pi) ft^2, we can set up the equation as follows:

(4/3)(pi)(r)^3 = 320(pi)

Simplifying the equation, we get:

(4/3)(r)^3 = 320

(r)^3 = (320 * 3) / 4

(r)^3 = 240

Taking the cube root of both sides, we get:

r = (240)^(1/3)

Therefore, the volume of the sphere is:

V = (4/3)(pi)(240)^(1/3) ft^3.

Answer 2

The volume of the sphere whose surface area is equal to the sum of the surface areas of the spheres with radii 4 ft and 8 ft is approximately 2381.36
`ft^3`.

How to find the volume of a sphere

To find the volume of a sphere whose surface area is equal to the sum of the surface areas of two spheres, determine the surface areas of the two spheres and then calculate the radius of the desired sphere.

The formula for the surface area of a sphere is given by:

A = 4π
`r^2`

For the first sphere with a radius of 4 ft, the surface area is:

A1 = 4π
`(4^2)`

A1 = 4π(16)

A1 = 64π
`ft^2`

For the second sphere with a radius of 8 ft, the surface area is:

A2 = 4π
`(8^2)`

A2 = 4π(64)

A2 = 256π
`ft^2`

The total surface area of the two spheres is the sum of A1 and A2:

A_total = A1 + A2

A_total = 64π + 256π

A_total = 320π ft^2

To find the radius of the desired sphere, we can rearrange the formula for the surface area of a sphere:

A = 4π
`r^2`

`r^2` = A / (4π)

`r^2` = 320π / (4π)

`r^2` = 80

Taking the square root of both sides

r = √80

r ≈ 8.944 ft

Now that we have the radius of the desired sphere, calculate its volume using the formula:

V = (4/3)π
`r^3`

V = (4/3)π
`(8.944^3)`

V ≈ 2381.36
`ft^3`

Therefore, the volume of the sphere whose surface area is equal to the sum of the surface areas of the spheres with radii 4 ft and 8 ft is approximately 2381.36
`ft^3`.