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5.1 Inverse & Joint Variation

5.1 Inverse & Joint Variation. p.303 What is direct variation? What is inverse variation? What is joint variation?. Just a reminder…. Direct Variation Use y=kx. Means “y varies directly with x.” k is called the constant of variation. New stuff!. Inverse Variation

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5.1 Inverse & Joint Variation

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  1. 5.1 Inverse & Joint Variation p.303 What is direct variation? What is inverse variation? What is joint variation?

  2. Just a reminder… Direct Variation Use y=kx. Means “y varies directly with x.” k is called the constant of variation.

  3. New stuff! Inverse Variation “y varies inversely with x.” k is the constant of variation.

  4. Ex: tell whether x & y show direct variation, inverse variation, or neither. • xy=4.8 • y=x+4 Inverse Variation Hint: Solve the equation for y and take notice of the relationship. Neither Direct Variation

  5. Ex: The variables x & y vary inversely, and y=8 when x=3. • Write an equation that relates x & y. k=24 • Find y when x= -4. y= -6

  6. a y= x a 3= 4 ANSWER 12 The inverse variation equation is y = x 12 = 6. Whenx = 2,y = 2 The variables xand yvary inversely. Use the given values to write an equation relating xand y. Then find ywhen x= 2. 4.x = 4,y = 3 Write general equation for inverse variation. Substitute 3 for yand 4 for x. 12 = a Solve for a.

  7. The number of songs that can be stored on an MP3 player varies inversely with the average size of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB). MP3 Players Write a model that gives the number nof songs that will fit on the MP3 player as a function of the average song size s(in megabytes).

  8. • Make a table showing the number of songs that will fit on the MP3 player if the average size of a song is 2MB, 2.5MB, 3MB, and 5MB as shown below. What happens to the number of songs as the average song size increases?

  9. STEP 2 Make a table of values. STEP 1 Write an inverse variation model. a n= s a 2500= 4 From the table, you can see that the number of songs that will fit on the MP3 player decreases as the average song size increases. ANSWER 10,000 s ANSWER A model is n = Write general equation for inverse variation. Substitute 2500 for n and 4 for s. 10,000 = a Solve for a.

  10. Computer Chips The table compares the area A(in square millimeters) of a computer chip with the number cof chips that can be obtained from a silicon wafer. • Write a model that gives cas a function of A. • Predict the number of chips per wafer when the area of a chip is 81 square millimeters.

  11. STEP 2 Make a prediction. The number of chips per wafer for a chip with an area of 81 square millimeters is STEP 1 Calculate the product A cfor each data pair in the table. c = 321 26,000 A c = 26,000 ,orc = 26,000 A 81 SOLUTION 58(448) = 25,984 62(424) = 26,288 66(392) = 25,872 70(376) = 26,320 Each product is approximately equal to 26,000. So, the data show inverse variation. A model relating Aand cis:

  12. Joint Variation • When a quantity varies directly as the product of 2 or more other quantities. • For example: if z varies jointly with x & y, then z=kxy. • Ex: if y varies inversely with the square of x, then y=k/x2. • Ex: if z varies directly with y and inversely with x, then z=ky/x.

  13. Examples: Write an equation. • y varies directly with x and inversely with z2. • y varies inversely with x3. • y varies directly with x2 and inversely with z. • z varies jointly with x2 and y. • y varies inversely with x and z.

  14. STEP 1 Write a general joint variation equation. STEP2 Use the given values of z, x, and y to find the constant of variation a. The variable zvaries jointly with xand y. Also, z= –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find zwhen x = 2 and y = 6. SOLUTION z = axy –75 = a(3)(–5) Substitute 75 for z, 3 for x, and 25 for y. Simplify. –75 = –15a 5 = a Solve for a.

  15. STEP 3 Rewrite the joint variation equation with the value of afrom Step 2. STEP 4 Calculate zwhen x = 2 and y = 6 using substitution. z = 5xy z = 5xy= 5(2)(6) = 60

  16. y = y = z = atr x = s a a ay x x2 x Write an equation for the given relationship. Relationship Equation a. yvaries inversely with x. b. zvaries jointly with x, y, and r. z = axyr c. y varies inversely with the square of x. d. zvaries directly with yand inversely with x. e. xvaries jointly with tand rand inversely with s.

  17. = a – 2 STEP 1 Write a general joint variation equation. STEP 2 Use the given values of z, x, and y to find the constant of variation a. 10.x = 4,y = –3,z =24 SOLUTION z = axy 24 = a(4)(– 3) Substitute 24 for z, 4 for x, and –3 for y. Simplify. 24 = –12a Solve for a.

  18. STEP 3 Rewrite the joint variation equation with the value of afrom Step 2. STEP 4 Calculate zwhen x = – 2 and y = 5 using substitution. ANSWER z = – 2 xy ; 20 z = – 2 xy z = – 2 xy= – 2 (– 2)(5) = 20

  19. What is direct variation? y varies directly with x (y = kx) • What is inverse variation? y varies inversely with x (y = k/x) • What is joint variation? A quantity varies directly as the product of two or more other quantities ( y = kxy)

  20. Assignment p. 307 3-33 every third problem,

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