4.1. Inverse Variation Models
www.ck12.org
4.1 Inverse Variation Models
Learning Objectives
•
•
•
•
Distinguish direct and inverse variation.
Graph inverse variation equations.
Write inverse variation equations.
Solve real-world problems using inverse variation equations.
Introduction
Many variables in real-world problems are related to each other by variations. A variation is an equation that relates
a variable to one or more variables by the operations of multiplication and division. There are three different kinds
of variation problems: direct variation, inverse variation and joint variation.
Distinguish Direct and Inverse Variation
In direct variation relationships, the related variables will either increase together or decrease together at a steady
rate. For instance, consider a person walking at a constant rate of three miles per hour. As time increases, the
distance covered by the person walking also increases at the rate of three miles each hour. The distance and time are
related to each other by a direct variation.
distance = rate × time
Since the speed is a constant 3 miles per hour, we can write: d = 3t.
Direct Variation
The general equation for a direct variation is of the form
y = kx.
k is called the constant of proportionality
You can see from the equation that a direct variation is a linear equation with a y−intercept of zero. The graph
of a direct variation relationship is a straight line passing through the origin whose slope is k the constant of
proportionality.
228
www.ck12.org
Chapter 4. Rational Equations and Functions
A second type of variation is inverse variation. When two quantities are related to each other inversely, as one
quantitiy increases, the other one decreases and vice-versa in a way that the product of the two quantities remains
constant.
For instance, if we look at the formula distance = rate × time again and solve for time, we obtain:
time =
distance
rate
If we keep the distance constant, we see that as the speed of an object increases, then the time it takes to cover
that distance decreases. Consider a car traveling a distance of 90 miles, then the formula relating time and speed is
t = 90
r .
Inverse Variation
The general equation for inverse variation is of the form
y=
k
x
where k is called the constant of proportionality.
In this chapter, we will investigate how the graph of these relationships behave.
Another type variation is a joint variation. In this type of relationship, one variable may vary as a product of two or
more variables.
For example, the volume of a cylinder is given by:
V = πr2 · h
In this formula, the volume varies directly as the product of the square of the radius of the base and the height of the
cylinder. The constant of proportionality here is the number π.
In many application problems, the relationship between the variables is a combination of variations. For instance
Newton’s Law of Gravitation states that the force of attraction between two spherical bodies varies jointly as the
masses of the objects and inversely as the square of the distance between them:
F =G
m1 m2
d2
229
4.1. Inverse Variation Models
www.ck12.org
In this example the constant of proportionality, G, is called the gravitational constant and its value is given by
G ≈ 6.674 × 10−11 N · m2 /kg2 .
Graph Inverse Variation Equations
We saw that the general equation for inverse variation is given by the formula y = kx , where k is a constant of
proportionality. We will now show how the graphs of such relationships behave. We start by making a table of
values. In most applications, x and y are positive. So in our table, we will choose only positive values of x.
Example 1
Graph an inverse variation relationship with the proportionality constant k = 1.
Solution
Since k = 1, the inverse variation is given by the equation y = 1x .
TABLE 4.1:
x
0
1
4
1
2
3
4
1
3
2
2
3
4
5
10
y
y=
y=
1
0
1
y=
1
y=
1
y=
y=
1
1
1
y=
y=
y=
y=
y=
1
2 = 0.5
1
3 ≈ 0.33
1
4 = 0.25
1
5 = 0.2
1
10 = 0.1
1
4
1
2
3
4
3
2
is undefined
=4
=2
≈ 1.33
=1
≈ 0.67
Here is a graph showing these points connected with a smooth curve.
Both the table and the graph demonstrate the relationship between variables in an inverse variation. As one variable
increases, the other variable decreases and vice-versa. Notice that when x = 0, the value of y is undefined. The graph
shows that when the value of x is very small, the value of y is very big and it approaches infinity as x gets closer and
closer to zero.
230
www.ck12.org
Chapter 4. Rational Equations and Functions
Similarly, as the value of x gets very large, the value of y gets smaller and smaller, but never reaches the value of
zero. We will investigate this behavior in detail throughout this chapter.
Write Inverse Variation Equations
As we saw earlier, an inverse variation fulfills the equation: y = kx . In general, we need to know the value of y at
a particular value of x in order to find the proportionality constant. After the proportionality constant is known, we
can find the value of y for any given value of x.
Example 2
y is inversely proportional to x, and y = 10 when x = 5. Find y when x = 2.
Solution
Since y is inversely proportional to x,
then the general relationship tells us:
Substitute in the values y = 10 and x = 5.
Solve for k by multiplying both sides of the equation by 5.
k
x
k
10 =
5
k = 50
y=
Now we put k back into the general equation.
The inverse relationship is given by:
When x = 2 :
50
x
50
or y = 25
y=
2
y=
Answer y = 25
Example 3
If p is inversely proportional to the square of q, and p = 64 when q = 3. Find p when q = 5.
Solution:
Since p is inversely proportional to q2 ,
then the general equation is:
Substitute in the values p = 64 and q = 3.
Solve for k by multiplying both sides of the equation by 9.
The inverse relationship is given by:
When q = 5 :
k
q2
k
k
64 = 2 or 64 =
3
9
k = 576
576
p= 2
q
576
p=
or p = 23.04
25
p=
Answer p = 23.04.
Solve Real-World Problems Using Inverse Variation Equations
Many formulas in physics are described by variations. In this section we will investigate some problems that are
described by inverse variations.
231
4.1. Inverse Variation Models
www.ck12.org
Example 4
The frequency, f , of sound varies inversely with wavelength, λ. A sound signal that has a wavelength of 34 meters
has a frequency of 10 hertz. What frequency does a sound signal of 120 meters have?
Solution
k
λ
k
10 =
34
k = 340
340
f=
λ
340
f=
⇒ f ≈ 2.83
120
f=
The inverse variation relationship is
Substitute in the values λ = 34 and f = 10.
Multiply both sides by 34.
Thus, the relationship is given by:
Plug in λ = 120 meters.
Answer f = 2.83 Hertz
Example 5
Electrostatic force
is the force of attraction or repulsion between two charges. The electrostatic force is given by the
formula: F = Kqd12q2 where q1 and q2 are the charges of the charged particles, d is the distance between the charges
and k is proportionality constant. In this example, the charges q1 and q2 do not change and are, thus, constants and
can then be combined with the other constant k to form a new constant K. The equation is rewritten as F = dK2 .
If the electrostatic force is 740 Newtons when the distance between charges is 5.3 × 10−11 meters, what is F when
d = 2.0 × 10−10 meters?
Solution
The inverse variation relationship is
F=
Plug in the values F = 740 and d = 5.3 × 10−11 .
740 =
Multiply both sides by (5.3 × 10−11 )2 .
K
(5.3 × 10−11 )2
2
K = 740 5.3 × 10−11 ≈ 2.08 × 10−18
The electrostatic force is given by
F=
When d = 2.0 × 10−10
F=
2.08 ∗ 10(−18)
Enter 2 into a calculator.
2.0 ∗ 10(−10)
K
d2
2.08 × 10−18
d2
2.08 × 10−18
(2.0 × 10−10 )2
F = 52
Answer F = 52 Newtons
Note: In the last example, you can also compute F =
232
2.08×10−18
2
(2.0×10−10 )
by hand.
www.ck12.org
Chapter 4. Rational Equations and Functions
F=
2.08 × 10−18
(2.0 × 10−10 )2
2.08 × 10−18
=
4.0 × 10−20
2.08 × 1020
=
4.0 × 1018
2.08 2 10
=
4.0
= 0.52(100)
= 52
This illustrates the usefulness of scientific notation.
Review Questions
Graph the following inverse variation relationships.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
y = 3x
y = 10
x
1
y = 4x
5
y = 6x
If z is inversely proportional to w, and z = 81 when w = 9, find w when z = 24.
If y is inversely proportional to x, and y = 2 when x = 8, find y when x = 12.
If a is inversely proportional to the square root of b, and a = 32 when b = 9, find b when a = 6.
If w is inversely proportional to the square of u and w = 4 when u = 2, find w when u = 8.
If x is proportional to y and inversely proportional to z, and x = 2 when y = 10 and z = 25, find x when y = 8
and z = 35.
If a varies directly with b and inversely with the square of c, and a = 10 when b = 5 and c = 2, find the value
of a when b = 3 and c = 6.
The intensity of light is inversely proportional to the square of the distance between the light source and the
object being illuminated. A light meter that is 10 meters from a light source registers 35 lux. What intensity
would it register 25 meters from the light source?
Ohm’s Law states that current flowing in a wire is inversely proportional to the resistance of the wire. If the
current is 2.5 Amperes when the resistance is 20 ohms, find the resistance when the current is 5 Amperes.
The volume of a gas varies directly to its temperature and inversely to its pressure. At 273 degrees Kelvin and
pressure of 2 atmospheres, the volume of the gas is 24 Liters. Find the volume of the gas when the temperature
is 220 dgreees Kelvin and the pressure is 1.2 atmospheres.
The volume of a square pyramid varies jointly as the height and the square of the length of the base. A square
pyramid whose height is 4 inches and whose base has a side length of 3 inches has a volume of 12 cubic
inches. Find the volume of a square pyramid that has a height of 9 inches and whose base has a side length of
5 inches.
Review Answers
1.
233
4.1. Inverse Variation Models
2.
3.
4.
234
www.ck12.org
www.ck12.org
5. w =
Chapter 4. Rational Equations and Functions
243
8
4
3
6. y =
7. b = 256
8. w =
9. x =
1
4
8
7
10. a =
2
3
11. I = 5.6 lux
12. R = 10 ohms
13. V ≈ 32.2 L
14. V = 75 in3
235