Question

The main diagonal of a rectangular prism is 31 units long and each dimension of the rectangular prism is an integer. What is the maximum and minimum possible volume of the rectangular prism?

(Working required)

Answer 1

There does not seem to any fromula to arrive at any result. The problem involves about finding three perfect square numbers which add up to 961.

If a,b and c are the dimensions ofthe rectangular prism, then main diagonal length d would be sqrt(a²+b²+c²) . That means, in the present case (31)² = a²+b²+c²

Since a,b and c are integers, the numbers a², b² and c² will all be perfect squares whose sum should be (31)² = 961

One such combination of perfect squares would be 900 +36+25, that means a=30, b=6, c=5. With these dimensions the volume would be 900 cubic units.

Another combination of perfect squares can be 441+324+196, that means a=21, b= 18 , c=14. With these dimensions the volume would be 5292 cubic units

This is then the required answer, unless some counter solution is given.