variation equations

7/28/2019 Variation Equations

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variables, they'll use the format in that third bulleted example above, or they'll say "varies as the sum

ofx andy.

Translatingvariation problems isn't so bad, once you get the hang of it. But then they want you to move

on to setting up and solving word problems. These generally fall into two categories: the ones where they

want you to find the value of "k", and the ones where they want you to find some other value, but only

after you've found "k" first. Here are some examples:

Ifyvaries directly as x2, and y= 8 when x= 2, find ywhen x= 1.

Since this is direct variation, the formula is "y = kx2". The reason they've given me the datapoint(x,y) = (2, 8) is that I have to be able to find the value of "k". So I'llplug inthe informationthey've given me, andsolvefork:

y = kx2

8 = k(22)8 = 4k2 = k

Now that I have k, I can rewrite the formula completely: y = 2x2. With this, I can answer thequestion they actually asked: "Findy whenx = 1."

y = 2x2y = 2(1)2y = 21

y = 2

Then the answer is: y= 2Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

Ifyvaries directly as xand z, and y= 5 when x= 3 and z= 4,

then find ywhen x= 2 and z= 3.

Translating the formula from English to math, I get:

y = kxz

Plugging inthe data point they gave me, andsolvingfor the value ofk, I get:

5 = k(3)(4)5 = 12k5/12 = k

Now that I have the value ofk, I can plug in the new values, and solve for the new value ofy:
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y = ( 5/12)xzy = ( 5/12)(2)(3)y = ( 5/12)(6)y = 5/2

Then the answer is: y=5

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Variat ion Equat ions(page 2 of 3)

Suppose that yis inversely proportional to x, and that y= 0.4 when x=

2.5. Find ywhen x= 4.

Translating, I get the formula:

y = k/x

Plugging in the data point they gave me, I can solve for the value of k:

y = k/x0.4 = k/(2.5)(0.4)(2.5) = k= 1

Now that I have found the value of the variation constant, I can plug in the x-value they gave me,and find the value ofy:

y = 1/x

y = 1/4

Then the answer is: y=1/4

Most word problems, of course, do not come all neatly arranged like the above examples. Instead, youhave to figure out what values go where.

According to Hooke's Law, the force needed to stretch a spring is proportional to theamount the spring is stretched. If fifty pounds of force stretches a spring five inches, how

much will the spring be stretched by a force of120 pounds?

"Is proportional to" means "varies directly with", so the formula for Hooke's Law is "F = kd",

where "F" is the force and "d" is the distance. (Note that, in physics, "weight" is a force. TheseHooke's Law problems are often stated in terms of weight, and the weight is the force.)

First I have to solve for the value ofk. They've given me the data point (d,F) = (5, 50), so I'llplug this in to the formula:


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F = kd50 = k510 = k

Now I know that the formula for this particular spring is "F= 10d". (Hooke's Law doesn't change,

but each spring is different, so each spring will have its own " k".) Once I know the formula, I cananswer their question: "How much will the spring be stretched by a force of 120 pounds?"

F= 10d120 = 10d12 = d

Note that they did not ask "What is the value of 'd'?". Be sure to answer the question they actuallyasked. The final answer is:

The spring will stretch twelve inches.

Kepler's third law of planetary motion states that the square of the time required for aplanet to make one revolution about the sun varies directly as the cube of the average

distance of the planet from the sun. If you assume that Mars is 1.5 times as far from the

sun as is the earth, find the approximate length of a Martian year.

This one is a bit different. The variation relationship is between the square of the time and thecube of the distance. The formula is:Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

t2 = kd3

If I take "d= 1" to mean "the distance is one AU", an AU being an "astronomical unit" (the

distance of earth from the sun), then the distance for Mars is 1.5 AU. Also, I will take "t= 1" tostand for "one earth year". Then, in terms of the planet Earth, I get:

(1)2 = k(1)31 = k

Then the formula, in terms of Earth, is:

t2 = d3

Now I'll plug in the information for Mars (in comparison to earth): d= 1.5:

t2 = (1.5)3

t=sqrt(3.375)

In other words, the Martian year is approximately the length of1.837 earth years (or just over

one year and ten months).

Variat ion Equat ions(page 3 of 3)


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The weight of a body varies inversely as the square of its distance from the center of the

earth. If the radius of the earth is 4000 miles, how much would a 200-pound man

weight 1000 miles above the surface of the earth?

Remembering that "weight" is a force, let the weight be designated by "F". The distance of a bodyfrom the center of the earth is "

d". Then the formula is the following:

F = k / d2

Plug in the given data point of(d,F) = (4000, 200) and solve fork:

200 = k/ (4000)2(200)(16,000,000) =k= 3,200,000,000

(Hey; there's nothing that says that khas to be small!)

Since the distance is always measured from the center of the earth, if the guy is in orbit a

thousand miles up, then his distance is the 4000 miles from the center to the surface plusthe1000 miles from the surface to his ship. That is, d= 5000. Plug this in, and solve forF:

F= (3,200,000,000) / (5000)2F= 128Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

Then the guy weighs 128 pounds.

Under certain conditions, the thrust Tof a propeller varies jointly as the fourth power of its

diameterdand the square of the numbernof revolutions per second. Show that, ifnis

doubled and dis halved, the thrust T is decreased by 75%.

Your first instinct is to say, "What the heck?", and your second is to say, "But they didn't give usany data points! We've got no numbers!". Here's a tip: when you have no idea what to do, tryplaying around with what they gave you, and see if anything useful happens.

At the very least, I can translate the formula from English into math:

T = k d4 n2

Now what? Well, whatever the diameter used to be, my new diameter is now half the olddiameter. And whatever the number of revolutions used to be, the new number is twice that

value. So I'll plug in "(

1

/2 )d= d/ 2" where "d" used to be, and plug in "2n" where "n" used tobe, and see if I can "find" the original "thrust" expression, k d4 n2, within the results:

new T= k( d/ 2 )4( 2n )2= k( d4 / 16 )( 4n2 )= k( d4 )( 1 / 16 )( 4 )( n2 )= k( d4n2 )( 4 / 16 )= ( kd4n2 )( 1 / 4 ) = ( 1 / 4 )( old T)


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In other words, when I make the changes they said to make, my new thrust is one-fourth of the

old thrust, which means that the thrust has been decreased by three-fourths, or75%.

The number of hours h that it takes mmen to assemble xmachines varies directly as thenumber of machines and inversely as the number of men. If four men can

assemble12 machines in four hours, how many men are needed to assemble 36 machines

in eight hours?

First, translate the English into math:

h = kx / m

Plug in the given data point and solve fork:

4 = k( 12 ) / ( 4 )16 = 12 k16/12 = k=

4/3

Now plug in the new information, and solve for the answer they want:

h = ( 4/3 )x / m8 = ( 4/3 )( 36 ) / m8 = 48 / m8m = 48m = 6

Remember that they didn't ask for the value of the variable m. I have to answer the question thatthey didask: "They will need six men."

As an aside, note that the language of variation is often used by pundits and politicians when they'retrying to sound smart, but they don't really know what the terms mean, so they use them incorrectly. Theywill say "this is directly proportional to that", when all they mean is that there is a positive statisticalcorrelation between the two things (increasing one thing makes the other thing increase, too). They willsay "this is inversely proportional to that", when all they mean is that there is a negative statisticalcorrelation between the two things (increase one thing makes the other thingdecrease). When listening topeople, keep in the back of your head what is the actual definition of the terms (in case you're listening to,say, a scientist, who will use the terms properly) and also the common misuse of the terms (in case you'relistening to, say, the evening news).

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