The length of the rectangular parallelepiped is 3 times its width and 4 times its height.

The length of the rectangular parallelepiped is 3 times its width and 4 times its height. Find the sum of the lengths of the edges of this parallelepiped if its surface area is 2250.

Let’s denote the width of the rectangular parallelepiped by x.

Since the length is 3 times the width, the length is 3x.

Since the length is 4 times less than the height, the height is 12x.

The surface area of ​​a rectangular parallelepiped is equal to the sum of the surface areas of its faces. In total, a rectangular parallelepiped has 6 faces, among which 3 pairs of equal faces can be distinguished. Their areas are equal to the product of length and height, height and width, length and width. Let’s compose and solve the equation:

2 * (x * 3x + x * 12x + 3x * 12x) = 2250;

(3 + 12 + 36) * x ^ 2 = 1125;

51 * x ^ 2 = 1125;

x ^ 2 = 375/17;

x = √ (375/17).

Let’s find the length and height:

3x = 3 * √ (375/17);

12x = 12 * √ (375/17).

In total, the parallelepiped has 12 edges, 4 of which are equal to the height, 4 are equal to the width and 4 are equal to the length. Let’s find their sum:

4 * (√ (375/17) + 3 * √ (375/17) + 12 * √ (375/17)) =
= 4 * 16 * √ (375/17) = 64 * √ (375/17) = 64 * √ (25 * 15/17) =
= 320 * √ (15/17).

Answer: 320 * √ (15/17).