Size - Square # 2 - Solution / Explanation

[Guest]

Here is a related problem to the previous:

A box with square base is to be constructed from a square piece of tin by removing 3 in. squares from the corners and folding up sides. Volume of box-48 in.^3. Determine the size piece of tin used.

The box has no top.

Partial solution provided:

Let X = unknown length of side.

Each side of base of box is X - 3 - 3 = X - 6.

Area of base is (X - 6)^2 and height is 3, volume is 3(X - 6)^2.

Volume is 3(X - 6) = 48 in.^3.

Solving for X:

(X - 6)^2 = 16 (dividing by 3)

X - 6 = + - 4 (taking square root)

X = 6 + - 4 (add 6)

X = 10
X = 2 (unacceptable)

I believe some steps are omitted. Please explain those steps. Thanks.

[Guest]

No steps are omitted. This is a complete solution with all the intermediate steps included.

There is however a small typo:
The line that says
Volume is 3(X - 6) = 48 in.^3.
should be
Volume is 3(X - 6)^2 = 48 in.^3.

Hope this helped,

R. Baber.

[Guest]

That helped again. Thanks.

One question:

"Area of base is (X - 6)^2"

Could the area be determined be determined by multiplying this out ?

[Guest]

Yes. The area is (X-6)^2=X^2-12X+36, the volume is therefore 3X^2-36X+108 = 48, which simplifies to 3X^2-36X+60=0 which simplifies to X^2-12X+20=0, factorizing the quadratic gives (X-10)(X-2)=0, giving the (same) two solutions X=10, and X=2 (only the former makes sense, given that the tin must be at least 3 by 3 inches so that 3 inch squares can be removed).

The method you use to simplify and solve equations never matters, provided you follow the rules you will always end up at the same correct answer. However, some methods are quicker and easier than others depending on the situation. In this case not expanding the area made solving for X a lot easier.

Hope this helped,

R. Baber.

[Guest]

That helped. Thanks.