1.
In a certain regular square pyramid, all of the edges have length 12. Find the volume of the pyramid.
2.
A rectangular prism has a total surface area of 56 Also, the sum of all the edges of the prism is 64. Find the length of the diagonal joining one corner of the prism to the opposite corner.
1. The volume of a square pyramid is given by the formula V=1/3*bh, where b is the length of the base and h is the height. In this case, the base is a square with side length 12, and the height is the distance from the apex of the pyramid to the center of the base. To find the height, we can use the Pythagorean theorem. The hypotenuse of the right triangle formed by the height, the slant height, and half the base is 12, and the slant height is 12. Therefore, the height is sqrt(12^2−(12/2)^2)=6*sqrt(2).
Plugging in b=12 and h=6*sqrt(2), we get V=1/3*(12)(6*sqrt(2))=24*sqrt(2).
2. Let the sides of the rectangular prism be l, b, and h. We know that the total surface area of the prism is 56, so we have:
2(lb+bh+lh)=56
We also know that the sum of all the edges of the prism is 64, so we have:
l+b+h=64
We can solve this system of equations to find that l=8, b=6, and h=4.
The length of the diagonal joining one corner of the prism to the opposite corner is given by:
sqrt(l^2+b^2+h^2)=sqrt(8^2+6^2+4^2)=sqrt(128)=8*sqrt(2)
Therefore, the length of the diagonal is 8*sqrt(2).
