SOLUTION: How many rectangular prisms with integer side lengths and exactly two square faces have equal volume and surface area?

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Question 1165991: How many rectangular prisms with integer side lengths and exactly two
square faces have equal volume and surface area?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The dimensions of the rectangular prism are x, x, and y.

The volume is x%5E2y

The surface area is 2x%5E2%2B4xy

We need to find the number of solutions in positive integers of

x%5E2y+=+2x%5E2%2B4xy

Divide out the common factor x. This is valid as long as x is not 0; and x=0 is not possible in this problem.

xy+=+2x%2B4y

Solve the equation for one variable in terms of the other.

xy-4y+=+2x
y%28x-4%29+=+2x
y+=+2x%2F%28x-4%29
y+=+%28%282x-8%29%2B8%29%2F%28x-4%29
y+=+%282x-8%29%2F%28x-4%29%2B8%2F%28x-4%29
y+=+2%2B8%2F%28x-4%29

In that final form, 2 and y are integers, so 8%2F%28x-4%29 has to be an integer.

The problem only asks for the number of solutions; the answer is the number of positive integral factors of 8, which is 4 (1, 2, 4,and 8).

The problem is more interesting if we continue to find the four solutions and verify that they are solutions.

Make a table showing the solutions and verifying that they are solutions, using the four positive integral factors of 8:
   x-4    x    8/(x-4)   y        x^2y          2x^2+4xy
 -----------------------------------------------------------------------
    1     5       8     10    25*10 = 250   2(25)+4(5)(10) = 50+200 = 250
    2     6       4      6    36*6 = 216    2(36)+4(6)(6) = 72+144 = 216
    4     8       2      4    64*4 = 256    2(64)+4(8)(4) = 128+128 = 256
    8    12       1      3    144*3 = 432   2(144)+4(12)(3) = 288+144 = 432