If we start with an 8 in. by 8 in. square of paper or cardboard, it can be folded into an open box (like a shoebox without the lid), if we first CUT AWAY equal-sided squares from the corners (see diagram).
(Math 8 through Pre-Calculus)
FOR MATH 8
If we start with an 8 in. by 8 in. square of paper or cardboard, it can be folded into an
open box (like a shoebox without the lid), if we first CUT AWAY equal-sided squares
from the corners (see diagram).
Our investigation will look at how the VOLUME
of the box is affected by "x" the length of the
side of the corner squares being removed before
the paper is folded.
If x is 0 in., there will be zero volume to the box - it will just be a flat piece of paper.
And if x is 4 in. the width and height will shrink to 0 in., leaving another zero volume
measure. These are the cases for boxes with the least or ",MINIMUM" volume. We'll
look at values BETWEEN 0 in. and 4 in. and try to find if there is a MAXIMUM volume
to the resulting boxes.
If we consider the smallest value of x to be .25 in. and the largest 3.75 in. (for
convenience) there will be 15 different multiples of .25 in., and 15 volumes.
You will be assigned TWO DIFFERENT VALUES OF X , .25 < X < 3.75 .
CALCULATE the volume of the boxes that will be produced. Make models of your
boxes from two sheets of paper.
A chart and graph will be created with the results from the entire class. This summary
will show how the volume is affected by our choice of x.
FOR SEQUENTIAL MATH 3 or PRE-CALCULUS
Create a VOLUME FUNCTION f(x) for the exercise described in terms of
"Y,"
the
length of the corner that's removed.
A. Use a spreadsheet to create a table of volumes for the various values of x. If there appears to be an interval with a MAXIMUM VOLUME, adjust the x inputs to get a "best value (within 0.1 cubic inches), or
B. Produce the curve of your f(x) by using a graphing calculator. Estimate the value of x which produces a maximum volume - adjust your domain (zoom in) to get your best value. NOTE: techniques will be developed in Calculus to determine the exact value of x which produces a maximum volume without trial-and-error or approximation.