PROBLEM:: I am given a rectangular sheet of cardboard 15 in. by
25 in. If a small square of the same size is cut from each corner and each
side folded along the cuts to form a lidless box, what size squares can
be cut from each corner to produce a box with a volume of 400 cubic in?
Also, what size square would produce the maximum volume?
One can see that the maximum volume of the box occurs when the value
of x=3. At this value for x, the box has a volume of 513 cubic inches. The
box dimensions would be 3 in. x 9 in. x 19 in.
To find the values of x when the volume equals 400 is rather simple now. By examining the spreadsheet, the volume 400 is found at approximately 1.5 and approximately 4.8. To narrow the value of x, I can adjust x on the spreadsheet. Now using, .01 as my increment for the x value, the volume is found precisely.
The volume of 400 is obviously when x = 1.53 and when x = 4.79. This
would make the dimensions of the box 1.53 in. x 11.94 in. x 21.94 in. or
4.79 in. x 5.42 in. x 15.42. in. ONe may wonder why there are two answers
to when the volume equals 400, and by looking at a graph of the function
explains this very clearly.
This graph reinforces what the spreadsheet told us, that the maximum
volume occurs at x =3 and the volume equals 400 at x=1.53 and x=4.79. The
graph is a parabola with a maximum value of x =3. The parabola crosses y=400
at two distinct locations, one on the increasing side of the parabola and
one on the decreasing side.
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