The two edges of a rectangular box going out from one vertex are 10 and 14

The two edges of a rectangular box going out from one vertex are 10 and 14. The surface area of this box is 568. Find the third edge going out of the same vertex.

1) We denote the area of the parallelepiped by S, and the lengths of its edges by a, b, c

Then, by the hypothesis of the problem, S = 568, a = 10, b = 14.

Let us derive a single formula for determining the length of the edge c for all four tasks.

The area of the parallelepiped S = 2 x (a x b + a x c + b x c).

S = 2 x a x b + 2 x a x c + 2 x b x c = 2 x a x b + c x (2 x a + 2 x b).

c = (S – 2 x a x b) / (2 x a + 2 x b) /

c = (568 – 2 x 10 x 14) / (2 x 10 + 2 x 14).

c = 288/48 = 6.

Answer: edge length c = 6.