The length of the diagonal of a regular quadrangular prism is 9cm. its total surface area is 144 cm ^ 2.

The length of the diagonal of a regular quadrangular prism is 9cm. its total surface area is 144 cm ^ 2. Find the side length of the base and the length of the side rib.

Let a regular quadrangular prism ABCDA1B1C1D1 be given. The length of the diagonal of the prism is DB1 = 9 cm, and its total surface area is Spol = 144 cm ^ 2. The length of the diagonal of the prism is found by the Pythagorean theorem through the diagonal of the base and the height DB1 ^ 2 = DB ^ 2 + BB1 ^ 2 = AB ^ 2 + BC ^ 2 + BB1 ^ 2. At the base of a regular quadrangular prism lies a square with side a, denote BB1 ​​= h, we get 2 ∙ a ^ 2 + h ^ 2 = 9 ^ 2. On the other hand, the total surface area of ​​a regular quadrangular prism is Spol = 2 ∙ a ^ 2 + 4 ∙ a ∙ h or 2 ∙ a ^ 2 + 4 ∙ a ∙ h = 144. Having solved the system of the two equations obtained, we find the length of the side a of the base and lateral rib length h. The problem has two solutions a1 = 4; a2 = 6, then h1 = 7; h2 = 3.