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Warm Up

Model Inverse and Joint Variation. Warm Up. Lesson Presentation. Lesson Quiz. y varies directly with x . If y = 36 when x = 8, find y when x = 5. 1. y = 22.5. ANSWER.

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Warm Up

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  1. Model Inverse and Joint Variation Warm Up Lesson Presentation Lesson Quiz

  2. yvaries directly with x. If y= 36 when x= 8, find y when x = 5. 1. y = 22.5 ANSWER If you travel at a constant speed, the distance you travel varies directly with time. If you travel 182 miles in 3.5 hours, how far will you travel at the same constant speed in 5 hours? 2. ANSWER 260 mi Warm-Up

  3. y = y c. = x 4 7 x Example 1 Tell whether xand yshow direct variation, inverse variation, or neither. Type of Variation Given Equation Rewritten Equation a.xy = 7 Inverse b.y = x + 3 Neither Direct y = 4x

  4. y= 7= ANSWER 28 The inverse variation equation is y = x 28 = –14. Whenx = –2, y = a a –2 x 4 Example 2 The variables xand yvary inversely, and y = 7 when x=4. Write an equation that relates xand y. Then find ywhen x = –2 . Write general equation for inverse variation. Substitute 7 for yand 4 for x. 28 = a Solve for a.

  5. Example 3 MP3Players 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). • 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).

  6. Example 3 • 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?

  7. STEP 1 Write an inverse variation model. a n= s a 2500= 4 ANSWER 10,000 s A model is n = Example 3 Write general equation for inverse variation. Substitute 2500 for n and 4 for s. 10,000 = a Solve for a.

  8. ANSWER 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. Example 3 STEP 2 Make a table of values.

  9. 0.75 y = x Guided Practice Tell whether xand yshow direct variation, inverse variation, or neither. Type of Variation Given Equation Rewritten Equation Direct 1. 3x = y y = 3x 2.xy = 0.75 Inverse Neither 3.y = x –5

  10. a y= x a 3= 4 ANSWER 12 The inverse variation equation is y = x 12 = 6. Whenx = 2,y = 2 Guided Practice 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.

  11. a y= x a –1= 8 ANSWER – 8 The inverse variation equation is y = x – 8 = – 4. Whenx = 2,y = 2 Guided Practice 5.x = 8,y = –1 Write general equation for inverse variation. Substitute –1 for yand 8 for x. – 8 = a Solve for a.

  12. , 6.x = y = 12 a y= x 1 a Substitute 12 for yand for x. 12= 1 2 2 ANSWER 6 The inverse variation equation is y = x 6 = 3. Whenx = 2,y = 2 1 2 Guided Practice Write general equation for inverse variation. 6 = a Solve for a.

  13. 7. What If? In Example 3, what is a model for the MP3 player if it stores 3000 songs when the average song size is 5MB? Write an inverse variation model. a n= s a 3000= 5 15,000 s ANSWER A model is n = Guided Practice Write general equation for inverse variation. Substitute 3000 for n and 5 for s. 15,000 = a Solve for a.

  14. Example 4 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.

  15. STEP 1 Calculate the product A cfor each data pair in the table. 26,000 A c = 26,000 ,orc = A Example 4 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:

  16. STEP 2 Make a prediction. The number of chips per wafer for a chip with an area of 81 square millimeters is c = 321 26,000 81 Example 4

  17. 26,000 26,000 329 = c = 79 A ANSWER about 329 chips Guided Practice 8. What If? In Example 4, predict the number of chips per wafer when the area of each chip is 79 square millimeters. SOLUTION The number of chips per wafer for a chip with an area of 79 square millimeters is

  18. 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. Example 5 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. –75 = –15a Simplify. 5 = a Solve for a.

  19. 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. Example 5 z = 5xy z = 5xy= 5(2)(6) = 60

  20. y = y = z = atr x = s ay a a x2 x x Example 6 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.

  21. STEP 1 Write a general joint variation equation. Guided Practice The variable zvaries jointly with xand y. Use the given values to write an equation relating x, y, and z. Then find zwhen x = –2 and y = 5. 9.x = 1,y = 2,z = 7 SOLUTION z = axy

  22. STEP 2 Use the given values of z, x, and y to find the constant of variation a. 7 = a 2 STEP 3 Rewrite the joint variation equation with the value of afrom Step 2. 7 z = xy 2 Guided Practice 7 = a(1)(2) Substitute 7 for z, 1 for x, and 2 for y. 7 = 2a Simplify. Solve for a.

  23. STEP 4 Calculate zwhen x = – 2 and y = 5 using substitution. 7 7 z = xy= (– 2)(5) = – 35 2 2 ANSWER ; – 35 7 z = xy 2 Guided Practice

  24. = 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. Guided Practice 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.

  25. 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 Guided Practice z = – 2 xy z = – 2 xy= – 2 (– 2)(5) = 20

  26. STEP 1 Write a general joint variation equation. Guided Practice The variable zvaries jointly with xand y. Use the given values to write an equation relating x, y, and z. Then find zwhen x = –2 and y = 5. 11.x = –2,y = 6,z = 18 SOLUTION z = axy

  27. STEP 2 Use the given values of z, x, and y to find the constant of variation a. 3 – = a 2 STEP 3 Rewrite the joint variation equation with the value of a from Step 2. 3 – z = xy 2 Guided Practice 18 = a(– 2)(6) Substitute 18 for z, –2 for x, and 6 for y. 18 = –12a Simplify. Solve for a.

  28. STEP 4 Calculate zwhen x = – 2 and y = 5 using substitution. – – 3 3 z = xy= (– 2)(5) = 15 2 2 ANSWER ; 15 3 – z = xy 2 Guided Practice

  29. STEP 1 Write a general joint variation equation. Guided Practice The variable zvaries jointly with xand y. Use the given values to write an equation relating x, y, and z. Then find zwhen x = –2 and y = 5. 12.x = –6,y = – 4,z = 56 SOLUTION z = axy

  30. STEP 2 Use the given values of z, x, and y to find the constant of variation a. 7 = a 3 STEP 3 Rewrite the joint variation equation with the value of a from Step 2. 7 z = xy 3 Guided Practice 56 = a(– 6)(–4) Substitute 56 for z, –6 for x, and –4 for y. 56 = 24a Simplify. Solve for a.

  31. STEP 4 Calculate zwhen x = – 2 and y = 5 using substitution. 7 7 70 70 – – z = xy= (– 2)(5) = 3 3 3 3 ANSWER ; 7 z = xy 3 Guided Practice

  32. a x w = y aqr p = s Guided Practice Write an equation for the given relationship. 13.xvaries inversely with yand directly with w. SOLUTION 14.pvaries jointly with qand r and inversely with s. SOLUTION

  33. – – 1. pand q vary inversely, and q = 15 whenp = 3.Writean equation that relatesp and q.Then findqwhenp = 45. ANSWER 45 ; q = 1 q = p 25amps 2. Write an equation for the following relationship: z varies jointly with the square of x and the cube root of y. ANSWER ANSWER z = ax2 y 3 Lesson Quiz 3. The current in a simple electrical circuit varies inversely with the resistance. If the current is 40amps when the resistance is 2.5ohms, find the current when the resistance is 4ohms.

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