A pyramid is inscribed in a cube whose volume is 52(pi) cm^3. The volume of a sphere is equal to the volume of this pyramid. Find the volume of a second cube whose edge is equal to the radius of the sphere.
V = (4pi/3)r^3 = 52picm^3.
Divide both sides by 4pi/3:
r^3 = 39,
r = 3.39cm.
V = (3.39)^3 = 39cm^3 = Volume of cube.
To find the volume of the second cube, we first need to find the volume of the sphere. Since the volume of the sphere is equal to the volume of the pyramid, we can find the volume of the sphere by finding the volume of the pyramid and then doubling it.
Given that the volume of the cube is 52π cm^3, we can find the length of the side of the cube using the formula:
Volume of a Cube = (Side length)^3
52π = (Side length)^3
Taking the cube root of both sides, we get:
Side length = ∛(52π)
Now let's find the volume of the pyramid. The volume of a pyramid can be found using the formula:
Volume of a Pyramid = (1/3) * Base Area * Height
In this case, the base of the pyramid is a square, and its side length is equal to the side length of the cube. So, the base area of the pyramid is:
Base Area = (Side length)^2
Substituting the value of the side length, we have:
Base Area = (∛(52π))^2
Next, we need to find the height of the pyramid. Since the pyramid is inscribed in the cube, the height of the pyramid is equal to the side length of the cube.
Height = Side length = ∛(52π)
Now, let's find the volume of the pyramid:
Volume of the Pyramid = (1/3) * Base Area * Height
= (1/3) * (∛(52π))^2 * ∛(52π)
Finally, to find the volume of the sphere, we double the volume of the pyramid:
Volume of Sphere = 2 * Volume of Pyramid
Now, we can find the volume of the second cube. Since its edge is equal to the radius of the sphere, we need to find the radius of the sphere first. The radius of the sphere is equal to the cube root of three times the volume of the sphere, divided by 4π:
Radius of Sphere = ∛((3 * Volume of Sphere) / (4π))
After finding the radius, we can calculate the volume of the second cube using the formula:
Volume of Cube = (Edge length)^3
Substituting the radius as the edge length, we have:
Volume of Cube = (Radius of Sphere)^3
By following these steps, you should be able to calculate the volume of the second cube.