Question

there's a pyramid inside a cube and the pyramid volume is 1944 what's the volume of the cube and length of the cube

Answer 1

The volume \( V \) of a pyramid can be calculated using the formula:

\[
V = \frac{1}{3} \times B \times h
\]

where \( B \) is the area of the base and \( h \) is the height of the pyramid.

For a pyramid inscribed in a cube, if we assume that the base of the pyramid is a square that lies on one face of the cube, and the apex (the top point) of the pyramid is at the opposite vertex of the cube, then the base of the pyramid will occupy one square face of the cube.

Now, let \( s \) be the length of a side of the cube. The base area \( B \) of the pyramid would be \( s^2 \), and the height of the pyramid \( h \) would also be \( s \) since it extends from the base to the opposite corner of the cube.

Therefore, the volume of the pyramid can be expressed as:

\[
V = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3
\]

Given that the volume of the pyramid is 1944, we can set up the equation:

\[
\frac{1}{3} s^3 = 1944
\]

Multiplying both sides by 3 gives:

\[
s^3 = 5822
\]

To find \( s \), take the cube root of both sides:

\[
s = \sqrt[3]{5822}
\]

Calculating this gives:

\[
s \approx 18.1
\]

Finally, to find the volume of the cube, we use the volume formula for the cube:

\[
\text{Volume of cube} = s^3 = 5822
\]

Thus, the volume of the cube is \( 5822 \) cubic units, and the length of each side of the cube is approximately \( 18.1 \) units.