M95. Mod 3. Sec. 6.5: Rational Equations. And Sec. 6.7 Variations:pg1

I. Solving Rational Equations: Be sure to consider the domain so you haven't introduced an extraneous solution when you multiply by a variable. This means you need to check your answers in the original equation each time.

Ex. 1:Domain:

LCD =

Ex. 2:Domain:

LCD =

M95, Sec. 6.5 & 6.7, pg 2

Ex. 3:Domain: Need to factor

LCD =

II. Introducing extraneous solutions:

Ex. 4:Domain:

LCD =

Ex. 5:Domain:

LCD =

M95, Sec. 6.5 & 6,7, pg 3

Ex. 6:Domain:

LCD =

Ex. 7:

M95, Sec. 6.5 & 6.7, pg 4

III. Using the negative to make factoring easier:

Ex. 8:

M95. Mod 4, Sec. 6.7: Proportions and Variations

I. A summary of some general variation models:

A. Direct Variation (direct proportion): Model:

We say: "y varies directly with x", "y and x vary directly" or "y is directly proportional to x". Note: y can also be proportional to , etc.

Proportion:

k:

B. Inverse Variation:Model:

We say: "x and y vary inversely" or "y is inversely proportional to x."

Note: y can also be inversely proportional to , etc.

M 95. Sec. 6.5 & 6.7, pg 5

C. Joint Variation:Model:

There will be more variables. We say: "y varies directly as the product of x and z" or "y varies jointly as x and z."

D. Combined Variation:Model:

There is more than one kind of variation going on. We say: "y varies directly as x and inversely as z."

II. Recognizing different types of proportions and variations.

For each of the following: a) Identify the type of variation: direct, inverse or joint b) find the constant of proportionality, k. and c) Write a phrase describing each variation (i.e. "y varies ...... as x").

Ex. 1:Ex. 2:

M 95. Sec. 6.5 &6.7, pg 6

Ex. 3:Ex. 4:

Ex. 5: Ex. 6:

III Examples using variations:

Steps to solve variation problems of any sort:

1. Write a general model using variables that describe the quantities. Choose the correct model based on the wording used. Be sure to include k in your model.

2. From the problem, fill in the variables and solve for k.

3. Write a specific model for your situation using the variables from #1 and the k-value found in #2.

4. Use the specific model and substitution to answer any questions.

Ex. 7:x and y vary directly. . Find x when y = 7.5

M 95. Sec. 6.5 &6.7, pg 7

Ex. 8: The speed of an auto in miles per hour (mph) varies directly with the speed in kilometers per hour (kmh). A speed of 64 mph is equivalent to a speed of 103 kmh. You are driving through Canada at 55 mph and see a sign that says the maximum speed is 80 kmh. Are you speeding?

Ex. 9: The distance a body falls varies directly as the square of the time of the fall. If a skydiver falls 64 feet in 2 seconds, how far will he fall in 3.5 seconds? 4.5 seconds? If he's dropped off at an altitude of 500 feet, how long will it take him to reach the ground?

Note: Always pay attention to units! They can keep you from going astray!

Ex. 10: Boyle's Law states that for a constant temperature, the pressure of a gas varies inversely with its volume. A sample of hydrogen gas has a volume of 8.56 liters at a pressure of 1.5 atmospheres. Find the volume of the hydrogen gas if the temperature remains constant and the pressure changes to 1.2 atmospheres.

M 95. Sec. 6.5 &6.7, pg 8

Ex. 11: The gravitational force with which earth attracts an object varies inversely as the square of the distance from the center of the earth. A gravitational force of 160 pounds acts on an object 400 miles from earth's center. Find the force of attraction on an object 6000 miles from the center of the earth.

Ex. 12: Much like the relationship between gravity and the distance of an object from the center of the earth, the intensity of a light source is inversely proportional to the square of the distance from the light source. A 100 watt light bulb has an intensity of 31.68 watts/square meter at a distance of half a meter. What would be the intensity of that bulb 2 meters away?

Ex. 13: The number of cars manufactured on an assembly line at a GM plant varies jointly as the number of workers and the time they work. If 200 workers can produce 60 cars in 2 hours, find how many cars 240 workers should be able to make in 3 hours.

Ex. 14: When a wind blows perpendicularly against a flat surface, its force is jointly proportional to the surface area and the speed of the wind. A sail whose surface area is 12 square feet experiences a 20-pound force when the wind speed is 10 miles per hour. Find the force on an 8-square foot sail if the wind speed is 12 miles per hour.