Section 1.3
Precalculus
Notes
Name:
Date:
Volume of a Box
Today we are going to explore how changing the dimensions on a rectangular sheet of paper, alters the
volume of the box it produces. Through this exploration, we will discuss the concept of maximum
volume as well as domain and range in real life problems.
x Please refer to the following diagram while completing this exploration:
11 – 2x x x 8.5 – 2x x Before you begin, make a prediction relating the height of the box and the volume that it will produce.
You will be assigned a “group” number that corresponds to the “group” column in the chart below. Your
“group” will be constructing a box with the provided dimensions, using an 8 ½’’ by 11’’ piece of paper.
As you are completing the construction phase of the box, please fill in the appropriate row in the chart.
We will discuss our findings as a class.
Group
Example
1
Dimensions of
square to cut out
of corners
0.5’’ by 0.5’’
1’’ by 1’’
2
1.5’’ by 1.5’’
3
2’’ by 2’’
4
2.5’’ by 2.5’’
Height of box
(x)
Length
(11 – 2x)
Width
(8.5 – 2x)
Volume
( v = lwh )
0.5’’
10’’
7.5’’
37.5 in 3
Discussion questions:
1.) Before we combine our results, which “groups” box do you think will have the largest volume?
2.) According to the chart, what is the maximum volume of a box that can be created from an 8 ½ ’’
by 11’’ sheet of paper? What height produced the maximum volume?
3.) Can we find a more exact value for the maximum volume of a box created from an 8 ½ ’’ by 11’’
sheet of paper? How?
4.) What equation would you use to represent the data?
5.) What would the graph of the equation you wrote in #3 look like (before you actually type it into a
graphing calculator!)? Is it realistic to look at the entire graph in this real life problem?
6.) Sketch a graph and find the exact value for the maximum volume of the box.
7.) At what height does the maximum occur?
8.) What is the domain of this real life problem?
9.) What is the range of this real life problem?
Homework: page 144 (#79) & Inverse Review Sheet
x 11 – 2x x x 8.5 – 2x
x