The diagonal of the rectangular parallelepiped, at the base of which the square lies, is equal to 3√3

The diagonal of the rectangular parallelepiped, at the base of which the square lies, is equal to 3√3, and the height takes on values belonging to the segment [1,5; 3.5]. Find the box with the largest volume.

We have a rectangular parallelepiped, at the base of which is a square.

The diagonal and the height of the parallelepiped are known. The height value is a range of values. Let’s find the maximum volume.

The volume of a parallelepiped is the product of the area of the base and the height. We find the area of the base from the Pythagorean theorem:

2 * a ^ 2 + h ^ 2 = d ^ 2;

2 * a ^ 2 = d ^ 2 – h ^ 2 = 27 – h ^ 2;

a ^ 2 = (27 – h ^ 2) / 2;

Volume formula:

V = a ^ 2 * h = 1/2 * (27 * h – h ^ 3).

The largest value of the volume is found as the largest value of the function.

Find the derivative of the volume:

V ‘= 1/2 * (27 – 3 * h ^ 2).

Let’s equate to zero:

27 – 3 * h ^ 2 = 0;

h = 3.

a ^ 2 = (27 – 9) / 2 = 9.

V = a ^ 2 * h = 27.