1. Warm Up
The length of the edge of a cube is 10
inches. How does the volume of a cube
with edges 3 times as long compare to the
volume of the smaller cube?

2. Warm Up
The length of the edge of a cube is 10
inches. How does the volume of a cube
with edges 3 times as long compare to the
volume of the smaller cube?
The volume of the large one is 27,000 cubic inches
while the smaller one is 1,000 cubic inches

3. Warm Up
The length of the edge of a cube is 10
inches. How does the volume of a cube
with edges 3 times as long compare to the
volume of the smaller cube?
The volume of the large one is 27,000 cubic inches
while the smaller one is 1,000 cubic inches
So, the volume will be 27 times as large.

4. 2.3 Fundamental
Theorem of
Variation &
2.9 Combined and
Joint Variation

5. THE ESSENTIAL
QUESTION
How do we solve variations?
What is the Fundamental Theorem of Variation?

6. The Fundamental
Theorem of Variation

7. The Fundamental
Theorem of Variation
If y varies directly as xn (y = kxn), and x
is multiplied by c, then y is multiplied by
cn.

8. The Fundamental
Theorem of Variation
If y varies directly as xn (y = kxn), and x
is multiplied by c, then y is multiplied by
cn.
If y varies inversely as xn (y = k/xn), and x
is multiplied by a nonzero constant c, then
y is divided by cn.

9. Suppose that the value of x is doubling, how is
y changing if:

10. Suppose that the value of x is doubling, how is
y changing if:
3
1. If y varies directly as x , then ______

11. Suppose that the value of x is doubling, how is
y changing if:
3
1. If y varies directly as x , then ______
y is multiplied by 8, from 23

12. Suppose that the value of x is doubling, how is
y changing if:
3
1. If y varies directly as x , then ______
y is multiplied by 8, from 23
2. If y varies directly as x4 , then ______

13. Suppose that the value of x is doubling, how is
y changing if:
3
1. If y varies directly as x , then ______
y is multiplied by 8, from 23
2. If y varies directly as x4 , then ______
y is multiplied by 16, from 24

14. Suppose that the value of x is doubling, how is
y changing if:
3
1. If y varies directly as x , then ______
y is multiplied by 8, from 23
2. If y varies directly as x4 , then ______
y is multiplied by 16, from 24
2
3. If y varies inversely as x , then ________

15. Suppose that the value of x is doubling, how is
y changing if:
3
1. If y varies directly as x , then ______
y is multiplied by 8, from 23
2. If y varies directly as x4 , then ______
y is multiplied by 16, from 24
2
3. If y varies inversely as x , then ________
y is divided by 4, from 22

17. 4. The formula I = k/D2 tells that the intensity of light varies
inversely as the square of the distance from the light source.
What effect does doubling the distance have on the intensity
of the light?

18. 4. The formula I = k/D2 tells that the intensity of light varies
inversely as the square of the distance from the light source.
What effect does doubling the distance have on the intensity
of the light?
We have an inverse variation so our
answer is divided by cn .

19. 4. The formula I = k/D2 tells that the intensity of light varies
inversely as the square of the distance from the light source.
What effect does doubling the distance have on the intensity
of the light?
We have an inverse variation so our
answer is divided by cn .
In the formula D is squared, so our n value
is 2, since we are doubling, our c value is 2.

20. 4. The formula I = k/D2 tells that the intensity of light varies
inversely as the square of the distance from the light source.
What effect does doubling the distance have on the intensity
of the light?
We have an inverse variation so our
answer is divided by cn .
In the formula D is squared, so our n value
is 2, since we are doubling, our c value is 2.
So we would divide by 4, which is the same as
multiplying by 1/4, so the light is 1/4 the intensity.

21. Combined & Joint Variation

22. Combined & Joint Variation
Combined variation - when direct and inverse
variations occur together

23. Combined & Joint Variation
Combined variation - when direct and inverse
variations occur together

24. Combined & Joint Variation
Combined variation - when direct and inverse
variations occur together
Joint variation - when one quantity varies
directly as the product of two or more
independent variables

25. Combined & Joint Variation
Combined variation - when direct and inverse
variations occur together
Joint variation - when one quantity varies
directly as the product of two or more
independent variables

26. Combined & Joint Variation
Combined variation - when direct and inverse
variations occur together
Joint variation - when one quantity varies
directly as the product of two or more
independent variables

27. Combined & Joint Variation
Combined variation - when direct and inverse
variations occur together
kx
for example - y=
z
Joint variation - when one quantity varies
directly as the product of two or more
€
independent variables

28. Combined & Joint Variation
Combined variation - when direct and inverse
variations occur together
kx
for example - y=
z
Joint variation - when one quantity varies
directly as the product of two or more
€
independent variables
for example - A = kbh

30. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.

31. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.

32. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
€

33. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?

34. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR
E=
I
€

35. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR 72k
E= 2.56 =
I 253
€ €

36. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR 72k 72k
E= 2.56 = 253⋅ 2.56 = ⋅ 253
I 253 253
€
€ €

37. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR 72k 72k
E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k
I 253 253
€ €
€ €

38. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR 72k 72k
E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9
I 253 253
€ €
€ € €

39. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR 72k 72k
E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9
I 253 253
9R
2.56 =
300
€ €
€ € €

40. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR 72k 72k
E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9
I 253 253
9R 9R
2.56 = 300 ⋅ 2.56 = ⋅ 300
300
€ 300 €
€ € €

41. 5. A baseball pitcher’s earned run average (ERA) varies directly
as the number of earned runs allowed and inversely as the
number of innings pitched. Write a general equation to model
this situation.
Let e = ERA, R = # of earned runs and I = # of innings.
kR
E=
I
6. In a recent year, a pitcher had an ERA of 2.56, having given up
72 earned runs in 253 innings. How many earned runs would the
pitcher have given up if he had pitched 300 innings, assuming
€
that his ERA remained the same?
kR 72k 72k
E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9
I 253 253
9R 9R
2.56 =
300
300 ⋅ 2.56 = ⋅ 300
300 €
R ≈ 85 runs
€ €
€ €

43. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.

44. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.

45. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r

46. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.

47. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first.

48. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first.
so... 1500 = k(15)2(10)

49. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first.
so... 1500 = k(15)2(10)
1500 = k(2250)

50. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first.
so... 1500 = k(15)2(10)
1500 = k(2250)
2/3 = k

51. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first. NOW: Find the power.
so... 1500 = k(15)2(10)
1500 = k(2250)
2/3 = k

52. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first. NOW: Find the power.
so... 1500 = k(15)2(10) P = (2/3)c2r
1500 = k(2250)
2/3 = k

53. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first. NOW: Find the power.
so... 1500 = k(15)2(10) P = (2/3)c2r
1500 = k(2250) P = (2/3)(20)2(21)
2/3 = k

54. 7. The power in an electrical circuit varies jointly as the square
of the current and the resistance. Write a formula to show this
relationship.
Let p = power, c = current and r = resistance.
Then P = kc2r
b. The power in a certain circuit is 1500 watts when the
current is 15 amps and the resistance is 10 ohms. Find
the power in that circuit when the current is 20 amps and
the resistance is 21 ohms.
HINT: Find k first. NOW: Find the power.
so... 1500 = k(15)2(10) P = (2/3)c2r
1500 = k(2250) P = (2/3)(20)2(21)
2/3 = k P = 5600 watts

55. Homework
worksheet 2.3A/2.9A