Comparing Dimensions
The first way the Subject Mathwill test your understanding of the relationship among
the basic measurements of geometric solids is by giving you the length, surface
area, or volume of different solids and asking you to compare their dimensions.
The math needed to answer comparing-dimensions questions isn’t that hard. But in
order to do the math, you need to have a good grasp of the formulas for each
type of solid and be able to relate those formulas to one another algebraically.
For example, The surface area of a sphere is the same as the volume of a cylinder. What is
the ratio of the radius of the sphere to the radius of the cylinder?
This question tells you that the surface area of a sphere and the volume a
cylinder are equal. A sphere’s surface area is 4p(
rs)2, where
rs is the radius
of the sphere.
A cylinder’s volume is p(rc)2 × h, where
rc is the radius
of the cylinder, and h is its
height. Therefore,
4p(
rs)
2 =p(
rc)
2 ×
h
The question asks for the ratio between the radii of the sphere and the
cylinder. This ratio is given by r s/rc.
Now you can solve the equation 4prs2
= prc2 × h for the ratio
rs/rc.
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Changing Measurements
The second way the Subject Mathwill test your understanding of the relationships
among length, surface area, and volume is by changing one of these measurements
by a given factor, and then asking how this change will influence the other
measurements.
When the lengths of a solid in the question are increased by a single constant
factor, a simple rule can help you find the answer:
- If a solid’s length is multiplied by a given factor, then the solid’s surface
area is multiplied by the square of that factor, and its volume is multiplied by
the cube of that factor.
Remember that this rule holds true only if all of a solid’s dimensions
increase in length by a given factor. So for a cube or a sphere, the rule holds
true when just a side or the radius changes, but for a rectangular solid,
cylinder, or other solid, all of the length dimensions must change by the same
factor. If the dimensions of the object do not increase by a constant factor—for
instance, if the height of a cylinder doubles but the radius of the base
triples—you will have to go back to the equation for the dimension you are
trying to determine and calculate by hand.
Example 1
If you double the length of the side of a square, by how much do you increase
the area of that square?
If you understand the formula for the area of a square, this question is simple.
The formula for the area of a square is A =
s2, where s is the
length of a side. Replace s with 2s, and you see that the area of a
square quadruples when the length of its sides double: (2s)2
= 4s2.
Example 2
If a sphere’s radius is halved, by what factor does its volume decrease?
The radius of the sphere is multiplied by a factor of 1/2
(or divided by a factor of 2), and so its volume multiplies by the cube of that
factor: (1/2)3 =
1/8. Therefore, the volume of the sphere is multiplied by
a factor of 1/8 (divided by 8), which is the same thing as
decreasing by a factor of 8.
Example 3
A rectangular solid has dimensions x × y ×
z (these are its length, width, and height), and a volume of
64. What is the volume of a rectangular solid of dimensions
x /2 × y /2 × z?
If this rectangular solid had dimensions that were all one-half as large as the
dimensions of the solid whose volume is 64, then its volume would be (1/2)3
× 64 = 1/8 × 64 = 8. But dimension z is not
multiplied by 1/2 like x and y. To answer a question like
this one, you should use the volume formula for rectangular solids: Volume = l
× w × h. It is given in the question that
xyz = 64. So, x/2 × y/2
× z = 1/4 × xyz = 1/4 × 64 = 16.
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