In a rectangular parallelepiped, the side edge is 12 cm, the area of the diagonal section is 312 cm²

In a rectangular parallelepiped, the side edge is 12 cm, the area of the diagonal section is 312 cm², and the area of the base of this parallelepiped is 240 cm². Calculate the sides of the base.

Since the area of the diagonal section is 312 cm ^ 2, the diagonal of the base is:

312: 12 = 26 cm.

We take the sides y of the base for x, then we get the system of equations:

x * y = 240;

x ^ 2 + y ^ 2 = 26 ^ 2.

Express x from the first and substitute it into the second:

x = 240 / y;

240 ^ 2 / y ^ 2 + y ^ 2 = 26 ^ 2;

y ^ 4 – 26 ^ 2 * y ^ 2 – 240 ^ 2 = 0;

y ^ 2 = 36;

y = 6.

Then:

x = 240: 6 = 40.

Answer: the sides of the base are 6 and 40 cm.