Presentation on theme: "5.1 Inverse & Joint Variation"— Presentation transcript:
15.1 Inverse & Joint Variation p.303What is direct variation?What is inverse variation?What is joint variation?
2Just a reminder… Direct Variation Use y=kx. Means “y varies directly with x.”k is called the constant of variation.
3New stuff! Inverse Variation “y varies inversely with x.” k is the constant of variation.
4Hint: Solve the equation for y and take notice of the relationship. Ex: tell whether x & y show direct variation, inverse variation, or neither.xy=4.8y=x+4Inverse VariationHint: Solve the equation for y and take notice of the relationship.NeitherDirect Variation
5Ex: The variables x & y vary inversely, and y=8 when x=3. Write an equation that relates x & y.k=24Find y when x= -4.y= -6
6The inverse variation equation is y = When x = 2, y = 2 = 6. ANSWER The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x = 2.4. x = 4, y = 3y =axWrite general equation for inverse variation.43 =aSubstitute 3 for y and 4 for x.12 = aSolve for a.12xThe inverse variation equation is y =When x = 2, y =2= 6.ANSWER
7MP3 PlayersThe 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 n of 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 • 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?
9Write an inverse variation model. a n = s a 2500 = 4 10,000 = a STEP 1Write an inverse variation model.an =sWrite general equation for inverse variation.a2500 =4Substitute 2500 for n and 4 for s.10,000 = aSolve for a.A model is n =s10,000ANSWERSTEP 2Make a table of values.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
10The table compares the area A (in square millimeters) of a computer chip with the number c of chips that can be obtained from a silicon wafer.Computer Chips• Write a model that gives c as a function of A.• Predict the number of chips per wafer when the area of a chip is 81 square millimeters.
11SOLUTIONSTEP 1Calculate the product A c for each datapair in the table.58(448) = 25,98462(424) = 26,28866(392) = 25,87270(376) = 26,320Each product is approximately equal to 26,000. So, the data show inverse variation. A model relating A and c is:A c = 26,000 , or c =A26,000STEP 2Make a prediction. The number of chips per wafer for a chip with an area of 81 square millimeters is8126,000321c =
12Joint VariationWhen 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.
13Examples: 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.
14Write a general joint variation equation. z = axy The variable z varies jointly with x and y. Also, z = –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find z when x = 2 and y = 6.SOLUTIONSTEP 1Write a general joint variation equation.z = axyUse the given values of z, x, and y to find theconstant of variation a.STEP 2–75 = a(3)(–5)Substitute 75 for z, 3 for x, and 25 for y.–75 = –15aSimplify.5 = aSolve for a.
15STEP 3Rewrite the joint variation equation with the valueof a from Step 2.z = 5xySTEP 4Calculate z when x = 2 and y = 6 using substitution.z = 5xy = 5(2)(6) = 60
16Write an equation for the given relationship. y =axa. y varies inversely with x.b. z varies jointly with x, y, and r.z = axyrc. y varies inversely with the square of x.y =ax2d. z varies directly with y and inversely with x.z =ayxe. x varies jointly with t and r and inversely with s.x =atrs
17Write a general joint variation equation. z = axy x = 4, y = –3, z =24SOLUTIONSTEP 1Write a general joint variation equation.z = axyUse the given values of z, x, and y to find the constant of variation a.STEP 224 = a(4)(– 3)Substitute 24 for z, 4 for x, and –3 for y.24 = –12aSimplify.= a– 2Solve for a.
18STEP 3Rewrite the joint variation equation with the value of a from Step 2.z = – 2 xySTEP 4Calculate z when x = – 2 and y = 5 using substitution.z = – 2 xy = – 2 (– 2)(5) = 20z = – 2 xy ; 20ANSWER
19What 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)