Algebra II
5.8 Launch Popcorn Problem
Name: _____________________________
Equal sized squares are cut from the corners of a 8.5 by 11 inch piece
of paper as shown at the right. An open box is formed by folding up
the sides (no top). Your goal is to build a box that can hold the most
popcorn possible. Your task is to determine how big the square should
be you that cut out of each corner to create the box with the greatest
volume.
1. If squares of side length 2 inches are removed from each corner of the
piece of paper, what are the height, length, width and volume of the box?
Height: _______
Length: _______
Width: _______
Volume: _______
2. If squares of side length 0.5 inches are removed from each corner of the piece of paper, what are the
height, length, width and volume of the box?
Height: _______
Length: _______
Width: _______
Volume: _______
3. Working with your group, complete the table below.
Length of
square removed
(inches)
Dimensions
(l, w, h)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Volume of the
box (cu inches)
4. From observing the table, what size square should you remove from each corner to make the biggest box?
5. If we instead let x represent the length of the side of the square we are removing from each corner, write
an expression to represent each of the following dimensions in terms of x.
Height: ________ Length: ________ Width: ________ Volume = _________________________
(leave in factored form)
6. Without expanding, determine the following information from the volume rule in #5:
Lead Term: _____ Degree: EVEN or ODD
“a” value: POS or NEG
End Behavior: __________
7. Find the zeros of the volume function algebraically
using the factored volume rule in #5. What do these
zeros tell you about the sheet of paper?
8. Sketch the graph without using a calculator by
plotting x-intercepts and using the end-behavior.
9. Highlight the portion of the curve on your graph above that makes sense for this problem. In other words,
which part of the cubic function represents dimensions of a box that you could actually make from a sheet
of paper?
10. Plot the point on your sketch that represents the greatest volume the box could have in real life. Use your
calculator to determine the coordinates of this point. What name do we give to this point on the graph?
Point: (__________,__________)
Name: ___________________________
Interpret both coordinates of this point in real-life context:
11. What are the dimensions of the biggest box you could make from the 8.5” by 11” sheet of paper? What is
the volume?
Dimensions: ________ by ________ by ________
Volume: _____________________
12.
Determine the practical domain and range for this particular problem based on your answers from #9-10.
D: ____________________________
R: ____________________________
**Now try to build this box from a piece of paper**