web viewthere is a relationship between the number cups of water wasted (w) and the number of...

Name _____________________________________________ Date ___________

Algebra2/Trig Apps: Direct VariationA dripping faucet wastes one cup of water for every three minutes it drips. 1. Use the table to show how much water is wasted as time passes.

# of minutes Cups of water wasted

3 1691230

2. There is a relationship between the number cups of water wasted (w) and the number of minutes (m) that passes. It can be expressed with a simple mathematical operation. Using an equation, or in words, express this relationship.

3. Explain, in words, why the dripping faucet example shows direct variation.

4. In the above situation, as time increases, the amount of water that has dripped increases/decreases (circle the correct word).

5. Fill out the y-values for the following table: x y

x=12

1234

6. Fill out the y-values for the following table: x xy=12123

Definition: Two variables are said to vary directly if their quotient is constant.

In this table, as x values increases, the y values increase/decrease.

Is xy=12 an example of direct variation? Why or why not?


Is yx =12 an example of direct variation? Why or why not?

Name _____________________________________________ Date ___________

47. Fill out the y-values for the following table:

x y = x + 3

1234

Writing equations of Direct Variation given initial conditions.Example: x and y vary directly. If x=12 when y=2, then what is x when y=24?Solution: In this question, you are not being asked if variables vary directly, you’re being TOLD they vary directly! Since they vary directly, that means that the quotient is constant. Set up an equation where you are showing that no matter what, the quotient is constant. This means that for any pair of values x and y, when you divide the values, the result will be the same.

Set up the equation: y1x1=y2x2

Replace the appropriate x and y values:Solve, by cross-multiplying.

Determining the Constant of Variation.The constant of variation, in this case, is the constant that you get when you divide the variables each time. What is the constant of variation in the above problem?


Is y = x + 3 an example of direct variation? Why or why not?

Name _____________________________________________ Date ___________

Algebra2/Trig Apps: Direct Variation HOMEWORKDetermine if each of the following represent DIRECT variation. If your answer is “yes,” determine the constant of variation.

a. 8

9

10

7

4

1

YX

b. c. d.

For each problem above, determine the constant of variation.Algebra2/Trig Apps: Inverse VariationA limo costs $800 to rent for the evening. The limo can fit a maximum of 10 couples. If each couple shares the cost of the limo, what is the cost per couple?

x y11 3312 3615 45

x y3 214 286 42

x y6 1810 3014 42

Name _____________________________________________ Date ___________

1. Fill out the table of values to show the cost per couple.# of couples Cost per couple

1 $80024810

2. There is a relationship between the number of couples (n) and the cost per couple (c). It can be expressed with a simple mathematical operation. Using an equation, or in words, express this relationship.

3. In the above situation, as the number of couples increase, the cost per couple increases/decreases (circle the correct word).

4. Fill out the y-values for the following table: x y

x=24

1234

5. Fill out the y-values for the following table: x xy=81248

6. Fill out the y-values for the following table: x y = 5 - x

Definition: Two variables are said to vary inversely if their product is constant.


Is xy=8 an example of inverse variation? Why or why not?


Is yx =24 an example of inverse variation? Why or why not?


Is y = 5 - x an example of direct variation? Why or why not?

Name _____________________________________________ Date ___________

1234

Writing equations of Inverse Variation given initial conditions.Example: x and y vary inversely. If x=12 when y=2, then what is x when y=24?Solution: In this question, you are not being asked if variables vary inversely, you’re being TOLD they vary inversely! Since they vary inversely, that means that the product is constant. Set up an equation where you are showing that no matter what, the quotient is constant. This means that for any pair of values x and y, when you divide the values, the result will be the same. Set up the equation: x1 ∙ y1=x2 ∙ y2Replace the appropriate x and y values:Solve.

Determining the Constant of Variation.The constant of variation, in this case, is the constant that you get when you multiply the variables each time. What is the constant of variation in the above problem?

Algebra2/Trig Apps: Inverse Variation HOMEWORKDetermine if each of the following represent INVERSE variation. If your answer is “yes,” determine the constant of variation.


Is y = 5 - x an example of direct variation? Why or why not?

x y0.5 6

1 3

Name _____________________________________________ Date ___________

1. 8

9

10

7

4

1

YX

2. 3. 4. 5

10

15

90

85

75

YX

For each of the above, determine the constant of variation.

x y

0.5 6

1 3

x y2 27

6 9

18 3

Name _____________________________________________ Date ___________

Algebra2/Trig Apps: Direct versus Inverse VariationWhich type of variation is represented by each of the following: inverse, direct, or neither? If you answer direct or inverse, write the equation.1. 2. 3.

4. 5. 6.

7. Heart rates and life spans of most mammals are inversely related. Mammal Heart rate in beats per

minuteLife span (in minutes)

Mouse 634 1,567,800Rabbit 158 6,307,200Lion 76 13,140,000Horse 63 15,768,000a. Use the data to write a function that models this inverse variation.

b. Use the equation you found in (a) to determine the approximate life span of a squirrel if you are given that its heart rate is 190 beats per minute.

c. Use the equation you found in (a) to determine the approximate heart rate of an elephant if you are given that its life span is about 70 years.

Algebra 2 Trig/Apps Homework

Name _____________________________________________ Date ___________

Suppose that x and y vary inversely. Write a function that models each inverse variation.

1. x = 1 when y = 11 2. x = - 13 when y = 100 3. x = 1 when y = 1

4. x = 28 when y = -2 5. x = 1.2 when y = 3 6. x = 2.5 when y = 100

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.

Suppose that x and y vary inversely. Write a function that models each inverse variation and find y when x = 10.

13. x = 20 when y = 5. 14. x = 20 when y = -4. 15. x = 5 when y = - 13

WARM –UP

Name _____________________________________________ Date ___________

Tell if the data has a direct variation relationship; if it is, give the constant of variation and writ the equation.

x y9 3

12 415 5

Joint Variation

Example 1

Suppose y varies jointly as x and z. Find y when x = 6 and z = 30 if y = 7 when z = 10 and x = 3.

Solution:

Use a proportion that relates the values.

y

(6)(30)= 7

(3)(10)

Cross Multiply

(7)(6)(30) = y(3)(10)

1260 = 30y

42 = y

Joint variation is the same as direct variation with two or more quantities. That is: Joint variation is a variation where a quantity varies directly as the product of two or more other quantities.

y = kxz or yxz=k

Name _____________________________________________ Date ___________

Your Turn

2. Assume a varies jointly with b and c. If b = 2 and c = 3, find the value of a. Given that a = 12 when b =1 and c = 6.

3. z varies jointly with x and y. Find z when x = 4 and y = 9 if x = 2 when y = 3 and z = 60.

4. Suppose c varies jointly with d and the square of g. c = 30 when d = 15 and g = 2. Find d when c = 6 and g = 8.

5. Suppose d varies jointly with r and t, and d = 110 when r = 55 and t = 2. Find r when d = 40 and t = 3.

Name _____________________________________________ Date ___________

ALGEBRA 2/TRIG APPS Mrs. von SteinHomework Joint Variation

1. If y varies jointly as x and z, and y =33 when x = 9 and z = 12, find y when x = 16 and z = 22.

2. If f varies jointly as g and the cube of h, and f = 200 when g = 5 and h = 4, find f when g= 3 and h = 6.

3. Wind resistance varies jointly as an object’s surface area and velocity. If an object traveling at 40 miles per hour with a surface area of 25 square feet experiences a wind resistance of 225 Newtons, how fast must a car with 40 square feet of surface area travel in order to experience a wind resistance of 270 Newtons?

4. For a given interest rate, simple interest varies jointly as principal and time. If $2000 left in an account for 4 years earns interest of $320, how much interest would be earned if you deposit $5000 for 7 years?

5. If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c = 36, find a when b = 9 and c = 225.

6. The volume of a pyramid varies jointly as its height and the area of its base. A pyramid with a height of 12 feet and a base with area of 23 square feet has a

Name _____________________________________________ Date ___________

volume of 92 cubic feet. Find the volume of a pyramid with a height of 17 feet and a base with an area of 27 square feet.

Name _____________________________________________ Date ___________

Name _____________________________________________ Date ___________

Algebra 2 Trig/Apps.SWBAT: Solve combined variation problemsWarm – Up1) The brightness of illumination, I, of an object varies inversely as the square of its

distance, d, from the source of illumination. If I = 18 luxes when d = 4m,

a. Find the value of k. _______________________

b. Find I when d = 3m._______________________

2) If d varies jointly as r and t, and d = 110 when r = 55 and t = 2, find r when d = 40 and t = 3.

_______________________

COMBINED VARIATION

Example 1y varies directly as x and inversely as z. If y = 5 when x = 3 and z = 4, find y when x = 6 and z = 8.

Example 2y varies directly as x2 and inversely as z. If y = 12 when x = 2 and z = 7, find y when x = 3 and z = 9.

Name _____________________________________________ Date ___________

YOUR TURN3. y varies directly as x and inversely as z. If y = 10 when x = 9 and z = 12, find y when x = 16 and z = 10.

4. x varies directly as y3 and inversely as z . If x = 7 when y = 2 and z = 4, find x when y = 3 and z = 9.

5. The number of girls varied directly as the number of boys and inversely as the number of teachers. When there were 50 girls, there were 20 teachers and 10 boys. How many boys were there when there were 10 girls and 100 teachers?

GRAPHING

Use the coordinate graphs to graph each of the following:6. xy = -20 7. yx =2

Name _____________________________________________ Date ___________

Algebra 2 Trig/Apps. COMBINED VARIATION HOMEWORK Due 9/26/13

1. Physics. The force F of gravity on a rocket varies directly with its mass m and inversely with

the square of its distance d from Earth. Write a model for this combined variation.

2. Horses varied directly as goats and inversely as pigs squared. When the barnyard contained

5 horses there were 4 pigs and only 2 goats. How many goats went with 6 pigs and 10 horses?

3. a) y varies jointly as x and w and inversely as the square of z. Find the equation of variation

when y = 100, x = 2, w= 4, and z= 20.

b) Then solve for y when x = 1, w =5 and z = 4.

Graph each of the following:

4. xy = 16 5. yx =−3

Name _____________________________________________ Date ___________

Name _____________________________________________ Date ___________

Algebra2/Trig Apps: Writing Equations of VariationSentence Variables needed Equation

(use “k” for the constant of variation.)

Extra Credit: What is the object and what is the value of the constant?

Example:The area of an object varies directly with the square of its radius,

A = area of the objectr = radius of the object

Ar2

=k

soA = kr2

Circlek = π

1. The area of an object varies jointly(directly) with the base and the height of the rectangle.

2. The volume of an object varies jointly with the area of its base and its height.

3. The volume of an object varies jointly with the square of its radius and its height.

4. The height of an object varies directly with its volume and inversely with the square of its radius.

5. The volume of an object varies jointly with its length, width, and height.

6. The length of an object varies directly with its volume, and inversely with the product of its width and height.

Name _____________________________________________ Date ___________

How to determine if a relationship is direct variation, inverse variation, or neither

Working with problems where it is GIVEN that a relationship is either direct variation or inverse

variation.

Read vars either left to right or top to bottom

both variables are INCreasing OR

both DECreasing

Possibly DIRECT variation.

Check to see if QUOTIENT of each x and

y is constant.

DIRECT VARIATIONx/y = k or

y/x = k

NEITHER

One var is INCreasing and the other is DECreasing

Possibly INVERSE variation.

Check to see if PRODUCT of each x and y is

constant.

INVERSE VARIATIONxy=k

NEITHER

Problem tells you it is DIRECT variation

DIVIDE y/x to determine the constant of

variation (k)equation is y/x=k

Problem tells you it is INVERSE variation

MULTIPLY xy to determine the constant

of variation (k)equation is xy=k

YES

NO

YES

NO

Name _____________________________________________ Date ___________

Test ReviewPart I: Am I inverse, direct, or neither?

a) For each of the following, indicate if the table of values shows inverse variation, direct variation, or neither.

b) If your answer is “inverse” or “direct,” write the exact equation (there should be no “k” in your equation.)

1. 2. 3. 4.

Part II: TranslationsTranslate the following sentences into equations using “k” to represent the constant of variation. (Each is an actual application of physics! )5.

6.

Part III: More Translations7. z varies jointly with x and y. a. Using a “k” to represent the constant, write the equation.

b. If x = 2 when y = 4 and z = 64, determine the value of k.

c. Using part a and b, determine the value of z when x = 4 and y = 10.

Name _____________________________________________ Date ___________

9. p varies directly with the square of x and inversely with y and z. a. Using a “k” to

represent the constant, write the equation.

b. If x = -4 when y = 2 , z= 10, and p = 40, determine the value of k.

c. Using part a and b, determine the value of p when x = -2, y = 6, and z = 1.

Part IV: Graphs10. Sketch the graph of xy= -4 on the axes below.

11. Sketch the graph of yx =−4 on the axes below.

12. Identify whether the following graphs are inverse or direct variation, and then write the equation.

Does this equation represent direct or inverse variation? How can you tell?

Does this equation represent direct or inverse variation? How can you tell?

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