The volume of a right trianglular prism is the area of the base times the height. The only case of this problem that can be solved is when the base is a right triangle. So the base must be a right triangle, hense it is a 3,4,5 right triangle. The area of the base is the product of the two legs divided by 2. In other words, 4*3/2 or 6. The volume is 6 times the height of 10 or 60.
To get 202.5 as the volume, we must multiply each side by a common factor. So the equation is (4x*3x)/2*10x = 202.5. This becomes 60x^3 = 202.5. Divide both sides by 60 to get x^3 = 3.375. The cube root of both sides yields x = 1.5. So the sides are 4.5, 6, 7.5, and 15.
To test multiply 4.5 * 6, divide by 2 and multiply the result by 15.
The volume of a right triangle prism is given as V = (a*b*h)/2, where b, c, d are the sides of the triangle, h is the height of the prism and a is the height of the triangle with sides b, c and d with b being the base.
Here, the ratio of the sides are 3:4:5:10.
If the triangle has the sides 3a, 4a and 5a, (3a)^2 + (4a^2) = (5a)^2; we see that if the base of the triangle is 4a the height is 3a.
Volume of the right triangle prism is (3a*4a*10a)/2 = 202.5
=> 60a^3 = 202.5
=> a^3 = 202.5/60
=> a = 1.5
The length of the longest side is 1.5*10 = 15