a Express the volume of the box in terms of x and use a calculator to draw a graph.
B
b Express the volume of the box in terms of x and use a calculator to draw a graph.
C
c Use the value of x you found in Part A.
A
a 2.94 inches
B
b 420 cubic inches
C
cLength: 14.12 inches
Width: 10.12 inches
Height: 2.94 inches
Practice makes perfect
a We start with a 16-inch by 20-inch piece of cardboard. Let's summarize the dimensions of the resulting box when we make cuts x inches long.
Dimension
Length
The length of the box is shorter than the length of the cardboard by twice the length of the cut.
l=20-2x
The width of the box is shorter than the width of the cardboard by twice the length of the cut.
w=16-2x
The height of the box is the length of the cut.
h=x
The volume of the box is the product of its length, width, and height.
V=(20-2x)(16-x)x
To find the greatest possible volume let's use a calculator to find the local maximum point on the graph of this function. We begin by pushing the Y= button and typing the equation of S in the first row.
To see the graph you will need to adjust the window. The maximum cut cannot be longer than half the width of the cardboard, so let's use 0 and 8 as horizontal bounds for the graph. Push WINDOW, change the settings, and push GRAPH.
To find the maximum point on the graph push 2nd and TRACE, then choose maximum from the menu. The calculator will prompt you to choose a left and right bound and to provide the calculator with a best guess of where the maximum might be.
The first coordinate of the maximum point gives the answer to this part of the question. A cut of about 2.94 inches will give a box with the greatest possible volume.
b The second coordinate of the maximum point we found in Part A gives the volume of the greatest possible box. The maximum volume we can get is about 420 cubic inches.
c Substituting x=2.94 in the expressions for the length, width, and height gives the dimensions of the finished box.
