The main diagonal of a rectangular prism is 31 units n length and each dimension of the rectangular prism is an integer. Find the maximum and minimum possible volume of the rectangular prism.
The only dimensions I can think of right off are 5,6,30
Using the following dinky 1980's style program in GW-BASIC
10 for a = 1 to 30
20 for b = 1 to 30
30 for c = 1 to 30
40 if sqr(a*a+b*b+c*c) = 31 then print a,b,c
50 next c
60 next b
70 next a
if got the following:
5,6,30 <------ Steve's answer
6,14,27
6,21,22
14,18,21
My little program actually gave me 24 answers, including all the permutations of the above four.
So now just multiply them to get the volume, and see which is smallest, and which is largest.
nicely done. Good luck finding those by hand.
To find the maximum and minimum possible volume of a rectangular prism, we first need to understand the properties of the main diagonal.
Let's assume the length, width, and height of the rectangular prism are represented by the variables x, y, and z, respectively.
Based on the Pythagorean theorem, we have the following equation:
x^2 + y^2 + z^2 = (31)^2
Since x, y, and z must all be integers, the main diagonal must also be an integer. Therefore, the value of (31)^2 must be the sum of three perfect squares of integers.
To find the maximum and minimum possible volume, we need to find the maximum and minimum possible values of x^2 + y^2 + z^2.
The maximum possible value for x^2 + y^2 + z^2 occurs when x, y, and z are as close to each other as possible. In this case, we would have x = y = z = 10 (approximately), yielding a volume of 10^3 = 1000 cubic units.
For the minimum possible value, we need to find the three smallest perfect squares whose sum equals (31)^2. This can be achieved by assigning the smallest possible values to x, y, and z.
To minimize the volume, we need to minimize the individual dimensions. Since the dimensions must be integers, we can start by setting x = 1, y = 2, and z = 5. This yields a volume of 1 * 2 * 5 = 10 cubic units.
Therefore, the maximum volume of the rectangular prism is 1000 cubic units, and the minimum volume is 10 cubic units.