Question

Johnny needs to make a rectangular box for his physics class project. He has bought P cm of wire and S cm2 of special paper. He would like to use all the wire (for the 12 edges) and paper (for the 6 sides) to make the box.

What is the largest volume of the box that Johnny can make?

Input

The first line contains t, the number of test cases (about 10). Then t test cases follow.
Each test case contains two integers P and S in a line (1 ≤ P ≤ 40000, 1 ≤ S ≤ 20000). You may assume that there always exists an optimal solution for the given input cases.

Output

For each test case, print a real number that is the largest volume of the box that Johnny can make, rounded to two decimal places.

Example Input:

2
20 14
20 16

Output:

3.00
4.15

Output details

First case: the dimensions of the largest box may be 3, 1 and 1.

Second case: the dimensions of the largest box may be 7/3, 4/3 and 4/3.

This is a practice problem from http://www.codechef.com. The name is “The Best box“. I don’t want the code for this. All I want to know is that how do we solve the problem? Any help would be appreciated. Thanks in advance.

Answer 1

You are actually trying to solve:

maximize V=a*b*c

subject to constraints:

4a+4b+4c = P 
2ab + 2ac + 2bc = S

This is a mathematical problem that can be solved using lagrange multipliers (Leaving developing the rest to you as an excercise – it is mostly technical, and if done slowly with care, shouldn’t be a problem).