Question

the volume of a pyramid that fits exactly inside a cube is 10 cubic feet. what is the volume of the cube?

Explanation:

Step1: Recall the volume formula for a pyramid

The volume formula for a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the base area and \(h\) is the height. When a pyramid fits exactly inside a cube, the base of the pyramid is a face of the cube and the height of the pyramid is equal to the side - length of the cube. Let the side - length of the cube be \(s\). Then the volume of the pyramid \(V_p=\frac{1}{3}s^2\times s=\frac{1}{3}s^3\), and the volume of the cube \(V_c = s^3\).

Step2: Use the given volume of the pyramid to find the volume of the cube

We know that \(V_p = 100\) cubic feet. Since \(V_p=\frac{1}{3}V_c\) (from the relationship \(V_p=\frac{1}{3}s^3\) and \(V_c = s^3\)), we can solve for \(V_c\) by using the formula \(V_c=3V_p\).
Substitute \(V_p = 100\) into the formula: \(V_c=3\times100\)

Answer:

300 cubic feet