4.12: Applications Using Direct Variation
Did you know that to find the flow rate of water in a pipeline, you can use the formula
Guidance
Direct variation has numerous realworld examples. You have already seen three examples: the area of a square is directly proportional to its side length; the distance you travel varies directly as the time you have been driving; and the total cost is directly proportional to the number of pounds of strawberries you purchase.
Newton's Second Law
In 1687, Sir Isaac Newton published the famous Principea Mathematica. It contained his Second Law of Motion. This law is often written as:
Acceleration is given in the units
Example A
If a 175 Newton force causes a heavily loaded shopping cart to accelerate down the aisle with an acceleration of
Solution: This question is basically asking us to solve for the constant of proportionality. Let us compare the two formulas.
We see that the two equations have the same form. The variable
Now solve for
Example B
If a 175 Newton force causes a heavilyloaded shopping cart to accelerate down the aisle with an acceleration of
Solution: Now you know the constant of variation is 70. In this formula, 70 represents the mass. To find the force needed to move the cart at an accelerated rate of 6 meters/second, substitute 6 for
When
The force needed to accelerate the cart is 420 Newtons.
Solving Direct Variation Using Proportions
You can use the Cross Products Theorem of proportions to solve direct variation situations. Because the fraction
Example C
Ohm’s Law states that the voltage
Suppose an electronics device passed a current of 1.3 amps at a voltage of 2.6 volts. What was the current when the voltage was increased to 12 volts?
Ohm’s Law matches with the definition of direct variation, so you can write a proportion.
Guided Practice
Suppose a car uses 5 liters of gasoline to drive 40 kilometers. How much gasoline will the car require in order to drive 128 kilometers?
Solution:
This is a problem of direct variation involving a constant that is kilometers per liter of gasoline. You can use a proportion to solve this problem. When setting up proportions, you need to make sure that the same unit is on the same side: top, bottom, left, or right. The same units cannot be placed so that they are on a diagonal from each other. It's easiest if you set up your proportion so that the unknown value is not in the denominator of the fraction. Since we are solving for liters, we will place liters in the numerators.
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Direct Variation Models (11:11)
 What two methods can be used to solve a direct variation problem?
 True or false? Every linear equation is a direct variation situation.
In 3 – 7, determine the constant of variation in each exercise.

y varies directly asx ; whenx=4,y=48 
d varies directly ast ; whent=7,d=329 
l varies directly ash ; whenl=112 ,h=−16 
m is directly proportional toh ; whenm=461.50 ,h=89.6 
z is directly proportional tor; whenr=412,z=51.5  Determine the equation of the strawberry purchase in the opener of this Concept.
 Dasan’s mom takes him to the video arcade for his birthday. In the first 10 minutes, he spends $3.50 playing games. If his allowance for the day is $20.00, how long can he keep playing games before his money is gone?
 The current standard for lowflow showerheads is 2.5 gallons per minute. Calculate how long it would take to fill a 30gallon bathtub using such a showerhead to supply the water.
 An electrical device passes a force of 288 volts at 32 amps. Using Ohm’s Law, determine:
 The constant of proportionality
 The force needed to pass 65 amps
 The diameter of a circle is directly proportional to its radius. If a circle with a 2inch diameter has a circumference of approximately 6.28 inches, what is the circumference of a 15inch circle?
 Amin is using a hose to fill his new swimming pool for the first time. He starts the hose at 10:00 P.M. and leaves it running all night. At 6:00 A.M. he measures the depth and calculates that the pool is foursevenths full. At what time will his new pool be full?
 Land in Wisconsin is for sale to property investors. A 232acre lot is listed for sale for $200,500. Assuming the same price per acre, how much would a 60acre lot sell for?
 The force
(F) needed to stretch a spring by a distancex is given by the equationF=k⋅x , where \begin{align*}k\end{align*} is the spring constant (measured in Newtons per centimeter, N/cm). If a 12Newton force stretches a certain spring by 10 cm, calculate: The spring constant, \begin{align*}k\end{align*}
 The force needed to stretch the spring by 7 cm
 The distance the spring would stretch with a 23Newton force
 Determine the equations of graphs \begin{align*}a  d\end{align*} below.
Image Attributions
Here you'll use various formulas to solve realworld direct variation problems.