aa section 2-9
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Combined and Joint VariationTRANSCRIPT
Section 2-9Combined and Joint Variation
Warm-upSolve for k.
1. 53 = k(2)(2)2
102. 80 =
k(3)(2)2
10
3. 107 =k(4)(2)2
104. 132.5 =
k(5)(2)2
10
Warm-upSolve for k.
1. 53 = k(2)(2)2
102. 80 =
k(3)(2)2
10
3. 107 =k(4)(2)2
104. 132.5 =
k(5)(2)2
10
k = 66.25
Warm-upSolve for k.
1. 53 = k(2)(2)2
102. 80 =
k(3)(2)2
10
3. 107 =k(4)(2)2
104. 132.5 =
k(5)(2)2
10
k = 66.25 k = 66 2/3
Warm-upSolve for k.
1. 53 = k(2)(2)2
102. 80 =
k(3)(2)2
10
3. 107 =k(4)(2)2
104. 132.5 =
k(5)(2)2
10
k = 66.25 k = 66 2/3
k = 66.875
Warm-upSolve for k.
1. 53 = k(2)(2)2
102. 80 =
k(3)(2)2
10
3. 107 =k(4)(2)2
104. 132.5 =
k(5)(2)2
10
k = 66.25 k = 66 2/3
k = 66.875 k = 66.25
QuestionWhat do you notice about the 4 warm-up problems?
What does it mean?
QuestionWhat do you notice about the 4 warm-up problems?
What does it mean?
More than one type of variation is going on!
Combined Variation
Combined Variation
When direct and inverse variation both occur in a problem
Combined Variation
When direct and inverse variation both occur in a problem
y =kxz
means y varies directly as x and inversely as z
Example 1A baseball pitcher’s ERA varies directly as the number of earned
runs allowed and inversely as the number of innings pitched. Write a model for the situation.
Example 1A baseball pitcher’s ERA varies directly as the number of earned
runs allowed and inversely as the number of innings pitched. Write a model for the situation.
E = ERA; R = runs; I = innings
Example 1A baseball pitcher’s ERA varies directly as the number of earned
runs allowed and inversely as the number of innings pitched. Write a model for the situation.
E = ERA; R = runs; I = innings
E =kRI
QuestionHow do you find k in a combined variation problem?
QuestionHow do you find k in a combined variation problem?
Keep all but one of the independent variables constant (like we saw in the warm-up). Choose a value for the other independent
variables and plug in.
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
25372(2.56) = k(72)
253i25372
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
25372(2.56) = k(72)
253i25372
k = 8.996
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
25372(2.56) = k(72)
253i25372
k = 8.996 E =8.996R
I
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
25372(2.56) = k(72)
253i25372
k = 8.996 E =8.996R
I2.56 = 8.996R
300
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
25372(2.56) = k(72)
253i25372
k = 8.996 E =8.996R
I2.56 = 8.996R
300
3008.996
(2.56) = 8.996R300
i3008.996
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
25372(2.56) = k(72)
253i25372
k = 8.996 E =8.996R
I2.56 = 8.996R
300
3008.996
(2.56) = 8.996R300
i3008.996
R = 85.37
Example 2Matt Mitarnowski had an ERA of 2.56, having given up 72
earned runs in 253 innings. How many earned runs would he have given up if he had pitched 300 innings if his ERA was the same?E =
kRI
2.56 = k(72)253
25372(2.56) = k(72)
253i25372
k = 8.996 E =8.996R
I2.56 = 8.996R
300
3008.996
(2.56) = 8.996R300
i3008.996
R = 85.37
Matt would have given up 85 runs
Joint Variation
Joint Variation
When the dependent variable varies directly as two or more independent variables, but there is no inverse variation
Joint Variation
When the dependent variable varies directly as two or more independent variables, but there is no inverse variation
y = kxz3 means y varies directly as x and the cube of z
QuestionHow do you find k in a joint variation problem?
QuestionHow do you find k in a joint variation problem?
The same as we did in a combined variation problem
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
P = kc2r
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
P = kc2r 1500 = k(15)2 (10)
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
P = kc2r 1500 = k(15)2 (10) k = 23
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
P = kc2r 1500 = k(15)2 (10) k = 23
P = 23c2r
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
P = kc2r 1500 = k(15)2 (10) k = 23
P = 23c2r P = 2
3 (20)2 (21)
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
P = kc2r 1500 = k(15)2 (10) k = 23
P = 23c2r P = 2
3 (20)2 (21) P = 5600
Example 3The power in an electric circuit varies jointly as the square of the
current and the resistance. The power is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power when the
current is 20 amps and the resistance is 21 ohms.P = Power; c = current; r = resistance
P = kc2r 1500 = k(15)2 (10) k = 23
P = 23c2r P = 2
3 (20)2 (21) P = 5600The power is 5600 watts
Homework
Homework
p. 125 #1 - 22