Download What is a direct variation?

Date: 5-7-12
Essential Question: What is a direct variation?
Topic: 8-9 Direct
Variation
Objective
To use direct variation to solve problems.
The table below shows the mass m of a gold bar whose volume
is V cubic centimeters.
Volume in cubic Mass in
centimeters: V
grams: m
Direct Variation
1
19.3
2
3
38.6
57.9
4
5
77.2
96.5
You can see that
. This equation defines a linear
function. Note that if the volume of the bar is doubled, the
mass is doubled; if the volume is tripled, the mass is tripled,
and so on. You can say that the mass varies directly as the
volume. This function is an example of a direct variation.
A direct variation is a function defined by an equation of
the form
𝑦 𝑘𝑥
where k is a nonzero constant.
You can say that y varies directly as x.
The constant k is called the constant of variation.
Summary
1
Example 1
State whether or not each equation defines a direct variation.
For each direct variation, state the constant of variation. If not
a direct variation, explain why.
a.
Yes
constant of variation = 3
b.
Yes
constant of variation = 5
c.
No
Domain variable (r) is squared.
d.
No
y-intercept = 2
2
Exercise 1
State whether or not each equation defines a direct variation.
For each direct variation, state the constant of variation. If not
a direct variation, explain why.
a.
b.
c.
d.
e.
3
Example 2
Given that m varies directly as n and that m = 42 when n = 2,
find the following:
a. the constant of variation
Let
.
Substitute m = 42 and n = 2
b. the value of m when n = 3
Let
.
Substitute k = 21 and n = 3
c. the value of n when m = 7
Let
.
Substitute k = 21 and m = 7
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Exercise 2
Given that v varies directly as u and that v = 12 when u = 6,
find the following:
a. the constant of variation
b. the value of v when u = 3
c. the value of u when v = 4
5
Constant of
Proportionality
) and (
) are two ordered pairs of a direct
Suppose (
variation defined by
and that neither x1 nor x2 is zero. Since
(
) and (
) must satisfy
, you know that
From these equations you can write the ratios
Since each ratio equals k. the ratios are equal.
This equation, which states that two ratios are equal, is a proportion.
For this reason, k is sometimes called the constant of proportionality,
and y is said to be directly proportional to x.
When you use a proportion to solve a problem, you will find it helpful
to recall that the product of the extremes equals the product of the
means. (See lesson 7-2.)
In the proportion a:b = c:d, a and d are called the
extremes, and b and c are called the means. You can
use the multiplication property of equality to show
that in any proportion the product of the extremes
equals the product of the means. That is:
𝑎
If 𝑏
𝑐
𝑑
, then ad = bc.
6
The amount of interest earned on savings is directly
proportional to the amount of money saved. If $104 interest is
earned on $1300, how much interest will be earned on $1800
in the same period of time?
Example 3
Step 1 The problem asks for the interest on $1800 if the interest on
$1300 is $104.
Step 2 Let i, in dollars, be the interest on d dollars.
i1 = 104
i2 = ? .
d1 = 1300
d2 = 1800
Step 3 An equation can be written in the form
Step 4
Step 5
 the interest earned on $1800 will be $144.
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Example 3
Solution 2
To solve Example 3 by the method shown in Example 2, first
write the equation
. Then solve for the constant of
variation, k, by using the fact that
when
.
Use the value of k to find the value of i when
.
Exercise 3
A certain car uses 15 gal of gasoline in 3 h. If the rate of
gasoline consumption is constant, how much gasoline will the
car use on a 35-hour trip?
8
Exercise 3 ct’d
A restaurant buys 20 lb of ground beef to prepare 110 servings
of chili. At this rate, how many servings can be made with
30 lb of ground beef?
Homework
P 393 Written Exercises: 1-21 every 4th
P 395 Problems: 1, 5, 9
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