171S2.6p Variation and Applications
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 2: More on Functions
2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry
2.5 Transformations 2.6 Variation and Applications
September 28, 2012
2.5 Variation and Applications
• Find equations of direct, inverse, and combined variation given values of the variables.
• Solve applied problems involving variation.
This is a good 9­minute Youtube video that relates to Section 2.6.
http://www.youtube.com/watch?v=is07Wg_0DiY Get Transformations Summary at http://cfcc.edu/faculty/cmoore/transformations­summary.pdf
Get Transformations xy tables at
http://cfcc.edu/faculty/cmoore/transformations­xy­tables.pdf Sep 22­2:20 PM
Direct Variation
If a situation gives rise to a linear function f(x) = kx, or y = kx, where k is a positive constant, we say that we have direct variation, or that y varies directly as x, or that y is directly proportional to x. The number k is called the variation constant, or constant of proportionality.
Direct Variation
The graph of y = kx, k > 0, always goes through the origin and rises from left to right. As x increases, y increases; that is, the function is increasing on the interval (0,∞). The constant k is also the slope of the line.
Sep 22­2:20 PM
Direct Variation
Example: Find the variation constant and an equation of variation in which y varies directly as x, and y = 42 when x = 3.
Solution: We know that (3, 42) is a solution of y = kx.
y = kx
42 = k × 3
14 = k
The variation constant 14, is the rate of change of y with respect to x. The equation of variation is y = 14x.
Application
Example: Wages. A cashier earns an hourly wage. If the cashier worked 18 hours and earned $168.30, how much will the cashier earn if she works 33 hours? Solution: We can express the amount of money earned as a function of the amount of hours worked.
I(h) = kh
I(18) = k × 18
$168.30 = k × 18
$9.35 = k The hourly wage is the variation constant.
Next, we use the equation to find how much the cashier will earn if she works 33 hours.
I(33) = $9.35(33)
= $308.55
Sep 22­2:20 PM
Inverse Variation
If a situation gives rise to a function f(x) = k/x, or y = k/x, where k is a positive constant, we say that we have inverse variation, or that y varies inversely as x, or that y is inversely proportional to x. The number k is called the variation constant, or constant of proportionality.
For the graph y = k/x, k > 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ∞).
Inverse Variation
For the graph y = k/x, k ≠ 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ∞).
Sep 22­2:20 PM
Inverse Variation
Example: Find the variation constant and an equation of variation in which y varies inversely as x, and y = 22 when x = 0.4.
Solution:
The variation constant is 8.8. The equation of variation is y = 8.8/x. Application
Example: Road Construction. The time t required to do a job varies inversely as the number of people P who work on the job (assuming that they all work at the same rate). If it takes 180 days for 12 workers to complete a job, how long will it take 15 workers to complete the same job?
Solution: We can express the amount of time required, in days, as a function of the number of people working. t varies inversely as P
This is the variation constant.
Sep 22­2:20 PM
Sep 22­2:20 PM
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171S2.6p Variation and Applications
September 28, 2012
Combined Variation
Application continued
The equation of variation is t(P) = 2160/P.
Next we compute t(15).
Other kinds of variation:
• y varies directly as the nth power of x if there is some positive constant k such that .
• y varies inversely as the nth power of x if there is some positive constant k such that .
• y varies jointly as x and z if there is some positive constant k such that y = kxz.
Example
It would take 144 days for 15 people to complete the same job.
The luminance of a light (E) varies directly with the intensity (I) of the light and inversely with the square distance (D) from the light. At a distance of 10 feet, a light meter reads 3 units for a 50­cd lamp. Find the luminance of a 27­cd lamp at a distance of 9 feet.
Solve for k.
Substitute the second set of data into the equation with the k value.
The lamp gives an luminance reading of 2 units.
Sep 22­2:20 PM
Find the variation constant and an equation of variation for the given situation.
218/2. y varies directly as x, and y = 0.1 when x = 0.2.
Find the variation constant and an equation of variation for the given situation.
218/4. y varies directly as x, and y = 12 when x = 5.
Sep 22­2:23 PM
219/16. Rate of Travel. The time t required to drive a fixed distance varies inversely as the speed r. It takes 5 hr at a speed of 80 km/h to drive a fixed distance. How long will it take to drive the same distance at a speed of 70 hm/h?
Sep 22­2:20 PM
Find the variation constant and an equation of variation for the given situation.
218/10. y varies inversely as x, and y = 1/5 when x = 35.
Find the variation constant and an equation of variation for the given situation.
218/12. y varies directly as x, and y = 0.9 when x = 0.4.
Sep 22­2:23 PM
219/20. Pumping Rate. The time t required to empty a tank varies inversely as the rate r of pumping. If a pump can empty a tank in 45 min at the rate of 600 kL / min, how long will it take the pump to empty the same tank at the rate of 1000 kL / min?
219/22. Relative Aperture. The relative aperture, or f­stop, of a 23.5­mm diameter lens is directly proportional to the focal length F of the lens. If a 150­mm focal length has an f­stop of 6.3, find the f­stop of a 23.5­mm diameter lens with a focal length of 80 mm. Sep 22­2:23 PM
Sep 22­2:23 PM
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171S2.6p Variation and Applications
September 28, 2012
220/24. Weight on Mars. The weight M of an object on Mars varies directly as its weight E on Earth. A person who weighs 95 lb on Earth weighs 38 lb on Mars. How much would a 100­lb person weigh on Mars? 220/28. Find an equation of variation for the given situation. y varies directly as the square of x, and y = 6 when x = 3.
220/26. Find an equation of variation for the given situation. y varies inversely as the square of x, and y = 6 when x = 3.
220/30. Find an equation of variation for the given situation. y varies directly as x and inversely as z, and y = 4 when x = 12 and z = 15.
Sep 22­2:23 PM
220/34. Find an equation of variation for the given situation. y varies jointly as x and z and inversely as the square of w, and y = 12/5, when x = 16, z = 3, and w = 5.
Sep 22­9:08 PM
Sep 22­9:05 PM
226/40. Boyles Law. The volume V of a given mass of a gas varies directly as the temperature T and inversely as the pressure P. If V = 231 cm3 when T = 42o and P = 20 kg/cm2, what is the volume when T = 30o and P = 15 kg/cm2? Sep 22­9:08 PM
226/40. Boyles Law. The volume V of a given mass of a gas varies directly as the temperature T and inversely as the pressure P. If V = 231 cm3 when T = 42o and P = 20 kg/cm2, what is the volume when T = 30o and P = 15 kg/cm2? Sep 22­9:08 PM
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