Greenwood press

VOLUME
159

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paper by measuring the height of a ream of paper and then dividing by 500
sheets. Hence, to find the volume of the sheet of paper, or the amount of wood
needed to make the paper, you would divide the volume of the prism formed by
the ream by the number of sheets of paper in the ream.
Manufactures think about volume as they build containers for their products.
Canned and boxed foods are often sold by their mass. Knowing the density of the
substance can help determine the amount of volume it will use in a container,
since density, 
d, is the ratio of mass, m, and volume, v. In terms of an equation, 
d =
m
v
. Nonuniform products that contain air pockets such as potato chips and 
cereal will often have additional empty volume when a package is opened,
because the contents will have settled and filled air pockets.
In addition to packaging food, companies that produce fragile items need to
consider the volume of additional materials that are needed, such as Styrofoam,
shredded paper, or packing bubbles. The amount of insulated packaging needed
would be equal to the difference between the volume of the box and the volume
of the item. If the item being shipped is in the form of a geometric solid, such as
a prism, pyramid, sphere, or cylinder, then the volume can be predicted with an
equation. For example, suppose a crystal ball with a radius of 2 inches is shipped
in a cubical container with an edge length of 6 inches. The volume of packaging
material needed to surround the crystal ball would be: the volume of the cube
minus the volume of the sphere 
= 6
3
−
4
3
π • 2
3
≈ 182 cubic inches. That is al-
most 85 percent of the space in the box!
Beverage production and distillation centers use the concept of volume to
determine how many containers can be filled based on their available raw mate-
rials. Cola companies need large tanks, usually cylindrical, to mix the raw ingre-
dients needed to create soft drinks. Once created, the cola will need to be emp-
tied into cans for distribution. Suppose a 5,000 gallon tank of cola is ready to be
dispersed into 12 ounce cans. If each gallon is equivalent to 128 fluid ounces,
then 5,000 
× 128 = 128,000 ounces of cola are available to produce a little more
than 53,000 soft drinks (128,000/12 
≈ 53,333), and over 2,200 (53,333/24 ≈
2,222) cases for distribution.
Ice cream cones are constructed so that the ice cream drips inside of the cone
as it melts. When ice cream is served, the spherical scoops lie on top of a cone
that is empty inside. The volume of ice cream inside the cone will gradually
increase as the temperature of the ice cream rises and pressure is applied at the
top of the cone. The cone keeps the ice cream inside it from melting more
quickly, since it is not exposed to the outside air temperature. An ice cream cone
with a height of 8 cm and base radius of 2 cm can hold close to half of a scoop
of ice cream with radius 2.5 cm. This is determined by dividing the volume of 
the cone 
1
3
π • 2
2
• 8 by the volume of the spherical scoop 
4
3
π • 2.5
3
, whose ratio
is approximately 0.512.
Construction workers who use concrete consider the amount of cement
needed to complete a job. When a driveway for a new house is planned, its