An ordinary sheet of paper is 8.5 inches by 11 inches. Find the volume of a cylinder when these values are treated as the height and circumference of the base.
B
b
Using the formula for the perimeter of a rectangle, start by computing what the height and circumference values must total to.
Comparison: The cylinder with the greater circumference has the greater volume.
B
b
A sheet of paper with dimensions 6.5 inches by 13 inches will roll into a right cylinder with the greatest volume.
Practice makes perfect
a
An ordinary sheet of paper is 8.5 inches by 11 inches. We can create two cylinders by alternating which side will be treated as the height of the cylinder and as the circumference of the base. Let's start by making h= 11 and C= 8.5. To find the volume we will use the formula for circumference to solve for the radius.
The volume of the cylinder when the shorter side of the paper is treated as the circumference is about 63.2 cubic inches. Let's repeat this process by making h= 8.5 and C= 11 this time. To find the volume we will use the formula for circumference to solve for the radius.
The volume of the cylinder when the longer side of the paper is treated as the circumference is about 81.8 cubic inches. If we compare the two volumes we can see that the cylinder with the greater circumference has the greater volume.
b
Let's start by recalling the formula for the perimeter of a rectangle.
A sheet of paper with perimeter of 39 inches must have the sum of its length and width equal 19.5. When creating a cylinder out of paper, one side of the paper will be the height and the other will be the circumference.
l+w=19.5
⇕
h+ C=19.5
⇕
C+ h=19.5
We will repeat the process in Part A with various combinations of height and circumference to determine which size can be rolled into a right cylinder with the greatest volume. We saw in Part A that the cylinder with the greater circumference has the greater volume, so we know the circumference must be at least 11 inches.
Height h
Circumference C
Radius found by r=C/2 π
V=Ï€ r^2 h
7.5
12
1.91
85.9
7.25
12.25
1.95
86.5
7
12.5
1.99
87.0
6.75
12.75
2.03
87.3
6.5
13
2.07
87.4
6.25
13.25
2.11
87.3
6
13.5
2.15
87.0
Looking at the table of values, we see that the volume is maximized when h= 6.5 and C= 13. In other words, of all sheets of paper with perimeter 39 inches (a 6.5 inches by 13 inches paper) can be rolled into a right cylinder with the greatest volume.
Checking Our Answer
Substituting More Values
To confirm that a 6.5 inches by 13 inches sheet of paper will create a cylinder with the greatest volume, we can check what the volume would be for height/ circumference values directly above and below 6.5 and 13.
Height h
Circumference C
Radius found by r=C/2 π
V=Ï€ r^2 h
6.49
13.01
2.071
87.3714
6.5
13
2.069
87.3715
6.51
12.99
2.067
87.3714
As we can see, the volume of the cylinder is maximized when the sheet of paper is 6.5 inches by 13 inches.
