### 171S2.6p Variation and Applications

```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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