Question 1027670: If the surface of a cube is 6x^2 -36x +54, what is the expression for the volume of the cube?
Found 3 solutions by mananth, Theo, ikleyn:
Answer by mananth(16949)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! formula for surface area of a cube is 6s^2.
s represents the length of the side of the cube.
that means that 6s^2 = 6x^2 - 36x + 54.
divide both sides of this equation by 6 and you get:
s^2 = x^2 - 6x + 9
factor that quadratic equation on the right side of the equation and you get:
s^2 = (x-3) * (x-3) = (x-3)^2
take the square root of both sides of the equation to get:
s = plus or minus (x-3).
since s can't be negative, then s = (x-3).
note that the value of x would have to be greater than 3, otherwise the length of the side would be negative.
the formula for the volume of a cube is s^3.
that means the volume of the cube must be equal to (x-3)^3.
since (x-3)^2 = x^2 - 6x + 9, then (x-3)^3 must be equal to:
(x-3) * (x^2 - 6x + 9) which would then be equal to x^3 - 9x^2 + 27x - 27.
how do you confirm this is correct?
assume the side of the cube is equal to 5 units.
the surface area would be 6s^2 = 6 * 5^2 = 6 * 25 = 150 square units.
the volume would be s^3 = 5^3 = 125 cubic units.
we had previously shown that s = x-3.
this means that x = s + 3.
if s = 5, this means that x = 8.
our formula for surface area is 6 * s^2 = 6.
since s = (x-3), then the surface area = 6 * (x-3)^2 = 6x^2 - 36x + 54.
replace x with 8 in these formulas and you will see that 6 * (x-3)^2 = 6 * 5^2 = 6 * 25 = 150 square units.
you will also see that 6x^2 - 36x + 54 = 6 * 8^2 - 36 * 8 + 54 = 384 - 288 + 54 = 150 square units.
the volume of the cube is equal to s^3.
when s = 5, this gets you the volume of the cube = 125 cubic units.
when s = 5, x = 8.
the volume of the cube = (x-3)^3 = (8-5)^3 = 5^3 = 125 cubic units.
the volume of the cube is also equal to x^3 - 9x^2 + 27x - 27.
when x = 8, this becomes (8)^3 - 9*8^2 + 27*8 - 27 which becomes equal to 512- 576 + 216 - 27 which becomes equal to 125 cubic units.
the formulas look good.
the expression for the volume of the cube would be (x-3)^3 or x^3 - 9x^2 + 27x - 27, depending on how you want to show it.
Answer by ikleyn(53742)  (Show Source):
You can put this solution on YOUR website! .
If the surface of a cube is 6x^2 -36x +54, what is the expression for the volume of the cube?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I will give different solution from that by @mananth and from that by @Theo,
and at the end I will explain why the solutions by both these tutors are incorrect and why they should be changed.
The surface area of the cube is given by the polynomial expression 6x^2 - 36x + 54.
It is the combined area of 6 identical faces of the cube.
The area of each face separately is
(6x^2 - 36x + 54)/6 = x^2 - 12x + 9 = (x-3)^2.
Notice that the given polynomial is always non-negative,
so its values make sense as the face area for all values of x =/= 3.
The length of the side of this cube is  = |x-3|.
Notice that I use the absolute value |x-3| here,
as the length is always assumed to be non-negative.
The volume of the cube is then  .
This formula works for all values x =/= 3, including x < 3,
while the formulas in the posts by @mananth and by @Theo do not work at x < 3 (!)
giving negative values for the edge length and for the volume.
So, my solution is MORE UNIVERSAL and more ACCURATE than the @mananth' solution and/or than the @Theo' solution.
Actually, this seemingly simple task has a   murderous  .
My solution discloses the trap, teaches you on how to avoid this trap
and also teaches you to be always aware and always accurate in Math.
------------------------------
It is I N T E R E S T I N G
Today, on Feb.14,2026, I submitted this problem to GOOGLE AI Overview to see how it treats it.
Google AI Overview practically repeated incorrect/incomplete solution by @mananth and by @Theo.
Naturally, I reported to Google AI about their fault through their feedback system.
Then I made my next experiment. I submitted the same problem to other AI,
math-gpt.org , which (i think) is slightly more advanced.
This other AI repeated the same tediousness and produced the same "lame" solution
(which reminds me a lame horse with three legs).
It is the real world of AI in the area of solving school Math problems, in which we all live now.
|
|
|
