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

Well, well, well, looks like we have a mathematical circus going on here! Let's jump through some hoops and solve this problem with a little laughter.

The surface area of a sphere is given by the formula A = 4πr², where r is the radius. So, let's calculate the surface areas of the two spheres:

For the first sphere with a radius of 4ft: A₁ = 4π(4)² = 16π ft²
And for the second sphere with a radius of 8ft: A₂ = 4π(8)² = 256π ft²

Now, let's find the surface area of the sphere we're looking for, which is equal to the sum of the surface areas of the previous two spheres:
A = A₁ + A₂ = 16π + 256π = 272π ft²

To find the radius of this mystery sphere, we can use the formula for surface area and rearrange it a bit:

A = 4πr²
272π = 4πr²
68 = r²
r ≈ ± 8.25

Uh-oh, we've got ourselves a little mathematical conundrum here! It seems like our solution has given us a negative value for the radius. Since we can't have a negative radius for a sphere, our calculations must have bounced a little too far.

But don't worry, the show must go on! Let's check our calculations and make sure we haven't dropped any laughter along the way.

To find the volume of a sphere whose surface area is equal to the sum of the surface areas of two spheres, we first need to find the surface areas of the two spheres.

The surface area of a sphere is given by the formula: A = 4πr^2, where A is the surface area and r is the radius.

For the first sphere with a radius of 4ft:
Surface area of first sphere = 4π(4^2)
= 4π(16)
= 64π ft^2

For the second sphere with a radius of 8ft:
Surface area of second sphere = 4π(8^2)
= 4π(64)
= 256π ft^2

Now, let's find the volume of the sphere whose surface area is equal to the sum of the surface areas of these two spheres.

The volume of a sphere is given by the formula: V = (4/3)πr^3, where V is the volume and r is the radius.

Let the radius of the third sphere be r3.
Given, surface area of the third sphere = surface area of the first sphere + surface area of the second sphere

4πr3^2 = 64π + 256π
r3^2 = (64 + 256)/4
r3^2 = 320/4
r3^2 = 80
r3 = sqrt(80)
r3 ≈ 8.94 ft

Now, we can find the volume of the third sphere using the formula:
Volume of the third sphere = (4/3)π(8.94^3)
≈ (4/3)(3.14)(8.94)^3
≈ 4.19 * 681.47
≈ 2851.99 ft^3

Therefore, the volume of the third sphere is approximately 2851.99 cubic feet.

To find the volume of a sphere, we need the formula for the volume of a sphere, which is V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.

First, let's find the surface area of each sphere. The formula for the surface area of a sphere is A = 4πr^2, where A is the surface area and r is the radius of the sphere.

For the sphere with a radius of 4ft, the surface area is A1 = 4π(4^2) = 4π(16) = 64π square feet.

For the sphere with a radius of 8ft, the surface area is A2 = 4π(8^2) = 4π(64) = 256π square feet.

Now, we need to find the surface area of the sphere whose surface area is equal to the sum of the surface areas of these two spheres. Let's denote the unknown radius of this sphere as r3.

We have the equation A3 = A1 + A2, where A3 is the surface area of the unknown sphere.

From here, we can substitute the formulas for the surface areas:

4πr3^2 = 64π + 256π

Dividing both sides of the equation by 4π, we get:

r3^2 = 16 + 64

r3^2 = 80

Taking the square root of both sides, we find:

r3 = √80

Simplifying the square root, we have:

r3 = 4√5

Now that we have the radius of the unknown sphere, we can calculate its volume. Using the formula V = (4/3)πr^3, we substitute r3 = 4√5:

V = (4/3)π(4√5)^3
= (4/3)π(4^3)(√5^3)
= (4/3)π(64)(5√5)
= (4/3)π(320√5)
= (2560π√5) cubic feet

Therefore, the volume of the sphere whose surface area is equal to the sum of the surface areas of the spheres with radii 4ft and 8ft is 2560π√5 cubic feet.

V = (4/3)pi r^3

A = 4 pi r^2

4 pi r^2 = 4 pi (16) + 4 pi (64)

r^2 = 80 = 16*5

r= 4 * sqrt 5

so
(4/3) pi [ 4^3 *5 sqrt 5 ]