Lesson 1
Let’s Make a Box
- Let’s investigate volumes of different boxes.
1.1: Which One Doesn’t Belong: Boxes
Which one doesn’t belong?
A.
length: 4 cm
width: 8 cm
height: 10 cm
B.
C.
D.
1.2: Building Boxes
Your teacher will give you some supplies.
- Construct an open-top box from a sheet of paper by cutting out a square from each corner and then folding up the sides.
- Calculate the volume of your box, and complete the table with your information.
side length of square cutout (in) | length (in) | width (in) | height (in) | volume of box (in3) |
---|---|---|---|---|
1 | ||||
1.3: Building the Biggest Box
-
The volume V(x) in cubic inches of the open-top box is a function of the side length x in inches of the square cutouts. Make a plan to figure out how to construct the box with the largest volume.
Pause here so your teacher can review your plan.
- Write an expression for V(x).
- Use graphing technology to create a graph representing V(x). Approximate the value of x that would allow you to construct an open-top box with the largest volume possible from one piece of paper.
The surface area A(x) in square inches of the open-top box is also a function of the side length x in inches of the square cutouts.
- Find one expression for A(x) by summing the area of the five faces of our open-top box.
- Find another expression for A(x) by subtracting the area of the cutouts from the area of the paper.
- Show algebraically that these two expressions are equivalent.
Summary
Polynomials can be used to model lots of situations. One example is to model the volume of a box created by removing squares from each corner of a rectangle of paper.
Let V(x) be the volume of the box in cubic inches where x is the side length in inches of each square removed from the four corners.
To define V using an expression, we can use the fact that the volume of a cube is (length)(width)(height). If the piece of paper we start with is 3 inches by 8 inches, then:
\displaystyle V(x) = (3-2x)(8-2x)(x)
What are some reasonable values for x? Cutting out squares with side lengths less than 0 inches doesn’t make sense, and similarly, we can’t cut out squares larger than 1.5 inches, since the short side of the paper is only 3 inches (since 3-1.5 \boldcdot 2=0). You may remember that the name for the set of all the input values that make sense to use with a function is the domain. Here, a reasonable domain is somewhere larger than 0 inches but less than 1.5 inches, depending on how well we can cut and fold!
By graphing this function, it is possible to find the maximum value within a specific domain. Here is a graph of y=V(x). It looks like the largest volume we can get for a box made this way from a 3 inch by 8 inch piece of paper is about 7.4 in3.
Glossary Entries
- polynomial
A polynomial function of x is a function given by a sum of terms, each of which is a constant times a whole number power of x. The word polynomial is used to refer both to the function and to the expression defining it.