If quantity y is directly related to quantity x and inversely related to z, what happens to y as x increases?   If quantity y is directly related to quantity x and inversely related to z, what happens to y as x decreases?

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Combined Variation

Combined variation involves a combination of direct or joint variation, and indirect variation. Since these equations are a little more complicated, you probably want to plug in all the variables, solve for k, and then solve back to get what’s missing.

Let’s try a problem:

Combined Variation Problem Math and Notes
(a)   y varies jointly as x and w and inversely as the square of z. Find the equation of variation when y=100, x=2, w=4, and z=20.

 

 

(b)  Then solve for y when x=1, w=5, and z=4.

ly1008kk=kxwz2=k(2)(4)(20)2=8k400=100(400)=(100)(400)8=5000                yyy=5000xwz2  (answer to a)=5000(1)(5)42=2500016=1562.5  (answer to b)

 

Now this looks really complicated, and you may get “word problems” like this, but all we do is fill in all the variables we know, and then solve for k. We know that “the square of z” is a fancy way of saying z2.

 

Remember that what follows the “varies jointly as” is typically on the top of any fraction (this is like a direct variation), and what follows “inversely as” is typically on the bottom of the fraction. And always put k on the top!

 

Now that we have the k, we have the answer to (a) above by plugging it in the original equation.

 

We can get the new y when we have “new” x, w, and z values.

 

For the second part of the problem, when x=1, w=5, and z=4, y=1562.5.   (Just plug in).

 

Combined Variation Word Problem:

Here’s another; this one looks really tough, but it’s really not that bad if you take it one step at a time:

Combined Variation Problem Math and Notes
The average number of phone calls per day between two cities has found to be jointly proportional to the populations of the cities, and inversely proportional to the square of the distance between the two cities.

 

 

The population of Charlotte is about 1,500,000 and the population of Nashville is about 1,200,000, and the distance between the two cities is about 400 miles. The average number of calls between the cities is about 200,000

 

(a)  Find the k and write the equation of variation.

 

(b)  The average number of daily phone calls between Charlotte and Indianapolis (which has a population of about 1,700,000) is about 134,000Find the distance between the two cities.

 

In reality, the distance between these two cities is 585.6 miles, so we weren’t too far off!

C200000kC=k(P1)(P2)d2=k(1500000)(1200000)4002=(200000)(400)2(1500000)(1200000)=.01778=.01778(P1)(P2)d2← answer to (a)

 

134000134000d2d=.01778(1500000)(1700000)d2=.01778(1500000)(1700000)=581.7miles← answer to (b)

 

We can set it up almost word for word from the word problem. Remember to put everything on top for “jointly proportional” (including k) since these are direct variations, and everything on bottom for “inversely proportional”.

 

Solve for k first; we get k=.01778.

 

Now we can plug in the new values to get the distance between the cities (d). We can actually cross multiply to get d2, and then take the positive square root get d.

 

The distance between Charlotte and Indianapolis is about 581.7 miles.

Questions:

  1. If quantity y is directly related to quantity x and inversely related to z, what happens to y as x increases?

 

  1. If quantity y is directly related to quantity x and inversely related to z, what happens to y as x decreases?

 

  1. What mathematical equations model combined variation?

 

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