A rectangular parallelepiped whose base is 12 by 20 is inscribed in a sphere of diameter 25. Find the volume of the part of the sphere outside of the parallelepiped. How would I do this, is the diagonal of the parallelepiped the diameter of the sphere? PLS HELP!!
Find the volume of the sphere (diameter 25) and subtract the volume of the parallelpipe. THis will give you the sphere outside the pipe : )
What would be the dimensions of the parallelepiped? That's what I don't understand, what would be the height?
The height is 25 (it is the same as the diameter : )
To find the volume of the part of the sphere outside the parallelepiped, we need to first determine the volume of the parallelepiped and the sphere.
The diagonal of the parallelepiped is not necessarily the diameter of the sphere in this case. However, we can use the diagonal length of the base (12, 20) to find the distance from the center of the base to any of its vertices, which will be the radius of the inscribed sphere.
Let's first find the volume of the parallelepiped. The volume of a parallelepiped is given by the product of its base area and height. Since the base is a rectangle with sides 12 and 20, the base area is given by A = length × width = 12 × 20.
Now, let's find the height of the parallelepiped. Since the parallelepiped is inscribed in the sphere, the height will be equal to the diameter of the sphere. Therefore, the height of the parallelepiped is 25.
The volume of the parallelepiped is given by V_parallelepiped = base area × height = 12 × 20 × 25.
Next, let's find the volume of the sphere. The volume of a sphere with radius r is given by V_sphere = (4/3)πr^3.
To find the radius of the inscribed sphere, we need to find the distance from the center of the base to any of its vertices (i.e., the diagonal length of the base). Using the Pythagorean theorem, the diagonal length is given by √(12^2 + 20^2).
Now, we can find the radius by dividing the diagonal length by 2: r = 1/2 * √(12^2 + 20^2).
Finally, substitute the value of the radius into the volume formula for the sphere to find V_sphere.
Once you have the volumes of the parallelepiped and the sphere, subtract the volume of the parallelepiped from the volume of the sphere to find the volume of the part of the sphere outside the parallelepiped.