find a rectangular solid such that total length = area =volume(without units)?

Question

Given dimensions a,b,c
Find a,b,c so that
4a +4b +4c = 2ab +2bc +2ac = abc

Answer 1

4a+4b+4c=2ab+2bc+2ac
4(a+b+c)=2(ab+bc+ac)
a=2
b=2
c=2

or
a=1
b=2
c=4

Answer 2

I break your equation into followings:
2*(a+b+c)=ab+ac+bc
a+b+c=abc/4
2*(ab+ac+bc)=abc
From the third equation, by dividing both sides to "abc" we have:
2*(1/a + 1/b + 1/c)=1 or 1/a+1/b+1/c=1/2
since all dimensions are positive numbers, the above equation requires a,b,c>=2
Putting this condition into the second equation, we find out the left side is at least equal to 6, but the right side is at least 2. However, it is obvious that by increasing any of dimensions, the right side grows more rapidly than the left side, and the limit is when a=b=c which requires 3*a=a^3/4 or a=2sqrt(3)
so 2 Assuming one dimension in this range will give two others by two equation solving. Suppose a=3:
3+b+c=3/4*bc
1/3+1/b+1/c=1/2
Solving above equations will give the other dimensions, and this will provide one set of solutions.

Answer 3

Fredy replies teh answer in best way!
Its an algebra exercise explaining possible solutions based on conditions.
Though 0 is one solution but its trivial!

But Fredy the method gives foll solution: (a=3)
b+c=6/7
bc=36/7

And if i am not wrong the values for b,c will be non real.Correct me if i am wrong!

Answer 4

I attempted to solve this ridiculous system of equations using numerical techniques. The one and only answer that ever came up was a=b=c=0. Essentially, it is impossible for this to happen in reality.

Answer 5

Single dimention math.
0

Answer 6

12a = 6 a^2

a = 2

6a^2 = a^3

a = 6

Answer 7

l=b=unity...
dis is possible if lenght n breadth is 1 metric unit...

do de other 1 urself...

Answer 8

FIRST OF ALL THERE IS NO SUCH TERM "RECTANGULAR SOLID".