SOLUTION: What is the length of the edge of a cube if after a slice 1cm thick is cut from one side, the volume remaining 294 cubic cm?

Question 1098180: What is the length of the edge of a cube if after a slice 1cm thick is cut from one side, the volume remaining 294 cubic cm?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
THE CARPENTER (OR THE CONFIDENT FIFTH-GRADER) SOLUTION:
I know how to calculate the volume of any shoe-box-shaped item.
You just multiply width, length and height.
For a cube, width length and height are all the same edge length.
If the cube edge length was 10 cm,
the initial cube volume would be
(10 cm) X (10 cm) X (10 cm) = 1000 cubic cm.
The volume of the slice cut off would be
(10 cm) X (10 cm) X (1 cm) = 100 cubic cm.
So, the final volume would be 900 cubic cm.
That is more than the 249 cm in the question.

I also know that the larger the original cube,
the larger the final volume,
and the larger the final volume required,
the larger the original cube needed.

What I know tells mne that the original cube edge must be less than 10 cm long.
It also tells me that there is only one answer.
If I calculate using increasing whole number lengths,
the calculated volumes will keep increasing,
so I will get to the answer and 249 cubic cm at some point,
or the answer was not a whole number length, and I will go past the answer.
I can calculate what the volume would be for a few edge lengths to see if I get 249 cubic cm as the answer.
If that happens, I will have the answer.
If not,
I will get less than 249 cubic cm for some whole number edge length,
but more than 249 cubic cm for the next whole number edge length,
and I will know that the answer is somewhere in between.

It is only common sense that the final volume is
more than the volume of a cube with edges 1 cm smaller.
I can easily calculate that a cube with edge length 5 cm has a volume of 125 cubic cm,
so the original cube's edge must be larger than 5 cm.

THE HIGH-SCHOOLER SOLUTION:
If
= length of the edge of the cube, in cm,
= volume of the cube in cubic cm, and
= volume of a slice 1cm thick is cut from one side of the cube.
So, is the volume remaining, in cubic cm.

All you have to do is solve <--> for .

Solving:

It so happens that
, so is a solution.

Is that the only solution?
The way the problwm is worded,
you would think that there is only one solution,
so answering ,
and going to the next problem would be a good strategy.

With time and willingless to spare, you could dig deeper
using whatever tools you have.

Using a graphing calculator, you could graph as
and find that is the only solution.

Using calculus:
could have 1 or 3 real zeros.
has zeros at and ,
representing respectively a local maximum and a local minimum for .
is the value of at its local maximum,
so to local maximum ,
decreases at to a local minimum at ,
and then increases for .
So, there can be only one real zero for ,
it happens for some ,
and as we already found that is a zero, we know know that is the only real zero for .
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