This document contains solutions to homework problems involving modeling with functions, volumes, surface areas, similar triangles, quadratic functions, and maximizing areas. The problems involve setting up and solving equations to find unknown variables in terms of other variables. The maximum area problems involve graphing functions and finding the vertex or local maximum.

1. 2.6 Modeling with Functions
Day 2

2. 2.6 Modeling with Functions
Day 2
Page 210 We will do these in class:
# 4, 8, 12, 16, 24, 28, 32

3. 2.6 Modeling with Functions
Day 2
Page 210 We will do these in class:
# 4, 8, 12, 16, 24, 28, 32
John 6:47 Truly, truly, I say to you, whoever believes has
eternal life.

4. 4)

5. 4)
2
V = πr h

6. 4)
2
V = πr h
h = 4r

7. 4)
2
V = πr h
h = 4r
2
V = π r ⋅ 4r

8. 4)
2
V = πr h
h = 4r
2
V = π r ⋅ 4r
3
V = 4π r

9. 8)

10. 8) Find SA in terms of V
x
x
x

11. 8) Find SA in terms of V
2
SA = 6x
x
x
x

12. 8) Find SA in terms of V
2
SA = 6x
x 3
V=x
x
x

13. 8) Find SA in terms of V
2
SA = 6x
x 3
V=x
x 1
x x =V 3

14. 8) Find SA in terms of V
2
SA = 6x
x 3
V=x
x 1
x x =V 3
1 2
⎛ ⎞
SA = 6 ⎜ V ⎟
3
⎝ ⎠

15. 8) Find SA in terms of V
2
SA = 6x
x 3
V=x
x 1
x x =V 3
1 2
⎛ ⎞
SA = 6 ⎜ V ⎟
3
⎝ ⎠
2
SA = 6V 3

16. 12)

17. 12)
Find L in terms of d
12
5
d L

18. 12)
Find L in terms of d
12
5
d L
These are similar
triangles so the sides
are proportional

19. 12)
Find L in terms of d
12
L L+d
5 =
5 12
d L
These are similar
triangles so the sides
are proportional

20. 12)
Find L in terms of d
12
L L+d
5 =
5 12
d L
12L = 5L + 5d
These are similar
triangles so the sides
are proportional

21. 12)
Find L in terms of d
12
L L+d
5 =
5 12
d L
12L = 5L + 5d
These are similar 7L = 5d
triangles so the sides
are proportional

22. 12)
Find L in terms of d
12
L L+d
5 =
5 12
d L
12L = 5L + 5d
These are similar 7L = 5d
triangles so the sides
are proportional 5d
L=
7

23. 16)

24. 16) Find P in terms of x
c
x
2x

25. 16) Find P in terms of x
c
x P = x + 2x + c
2x

26. 16) Find P in terms of x
c
x P = x + 2x + c
2 2 2
c = x + 4x
2x

27. 16) Find P in terms of x
c
x P = x + 2x + c
2 2 2
c = x + 4x
2x
2
c = 5x

28. 16) Find P in terms of x
c
x P = x + 2x + c
2 2 2
c = x + 4x
2x
2
c = 5x
c=x 5

29. 16) Find P in terms of x
c
x P = x + 2x + c
2 2 2
c = x + 4x
2x
2
c = 5x
c=x 5
P = 3x + x 5

30. 16) Find P in terms of x
c
x P = x + 2x + c
2 2 2
c = x + 4x
2x
2
c = 5x
c=x 5
P = 3x + x 5
(
P = 3+ 5 x )

31. 24)

32. 24) y
x x x x x
y

33. 24) y
a) Find Area in terms of x
x x x x x
y

34. 24) y
a) Find Area in terms of x
x x x x x
y
5x + 2y = 750

35. 24) y
a) Find Area in terms of x
x x x x x
y
5x + 2y = 750
750 − 5x
y=
2

36. 24) y
a) Find Area in terms of x
x x x x x
A = lw
y
5x + 2y = 750
750 − 5x
y=
2

37. 24) y
a) Find Area in terms of x
x x x x x
A = lw
y ⎛ 750 − 5x ⎞
A = ( x ) ⎜ ⎟
5x + 2y = 750 ⎝ 2 ⎠
750 − 5x
y=
2

38. 24) y
a) Find Area in terms of x
x x x x x
A = lw
y ⎛ 750 − 5x ⎞
A = ( x ) ⎜ ⎟
5x + 2y = 750 ⎝ 2 ⎠
750 − 5x
y= 750x − 5x 2
2 A=
2

39. 24) y
b) Find the Maximum Area
x x x x x graph and find the max
y
2
750x − 5x
A=
2

40. 24) y
b) Find the Maximum Area
x x x x x graph and find the max
2
750x − 5x
y y1 =
2
2
750x − 5x
A=
2

41. 24) y
b) Find the Maximum Area
x x x x x graph and find the max
2
750x − 5x
y y1 =
2
2
750x − 5x
A= vertex : ( 75,14062.5 )
2
in the form of (x,A)

42. 24) y
b) Find the Maximum Area
x x x x x graph and find the max
2
750x − 5x
y y1 =
2
2
750x − 5x
A= vertex : ( 75,14062.5 )
2
in the form of (x,A)
max area is 14,062.5 sq. ft.

43. 28)

44. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses

45. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20

46. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20
let n = the number of $1 increases

47. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20
let n = the number of $1 increases
price per item: 10+n # sold: 20-2n

48. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20
let n = the number of $1 increases
price per item: 10+n # sold: 20-2n
∴ P = (20 − 2n)(10 + n) − (20 − 2n)(6)

49. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20
let n = the number of $1 increases
price per item: 10+n # sold: 20-2n
∴ P = (20 − 2n)(10 + n) − (20 − 2n)(6)
b) find the maximum profit

50. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20
let n = the number of $1 increases
price per item: 10+n # sold: 20-2n
∴ P = (20 − 2n)(10 + n) − (20 − 2n)(6)
b) find the maximum profit
graph and find the max (vertex)

51. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20
let n = the number of $1 increases
price per item: 10+n # sold: 20-2n
∴ P = (20 − 2n)(10 + n) − (20 − 2n)(6)
b) find the maximum profit
graph and find the max (vertex)
vertex : ( 3,98 ) as ( n, P )

52. 28) a) profit = (# sold)(price each)-(# sold)(cost each)
profit = revenue - expenses
Given: price per item: $10 # sold: 20
let n = the number of $1 increases
price per item: 10+n # sold: 20-2n
∴ P = (20 − 2n)(10 + n) − (20 − 2n)(6)
b) find the maximum profit
graph and find the max (vertex)
vertex : ( 3,98 ) as ( n, P )
max profit of $98 occurs when the price is set at $7 each

53. 32)

54. 32) Maximize the area
(x,y) of the rectangle
y y
x x
2
y= 8− x

55. 32) Maximize the area
(x,y) of the rectangle
A = (2x)(y)
y y
x x
2
y= 8− x

56. 32) Maximize the area
(x,y) of the rectangle
A = (2x)(y)
y y
2
x x
y= 8− x
2
y= 8− x

57. 32) Maximize the area
(x,y) of the rectangle
A = (2x)(y)
y y
2
x x
y= 8− x
2
2 A = (2x)(8 − x )
y= 8− x

58. 32) Maximize the area
(x,y) of the rectangle
A = (2x)(y)
y y
2
x x
y= 8− x
2
2 A = (2x)(8 − x )
y= 8− x
graph this cubic and look for a local max

59. 32) Maximize the area
(x,y) of the rectangle
A = (2x)(y)
y y
2
x x
y= 8− x
2
2 A = (2x)(8 − x )
y= 8− x
graph this cubic and look for a local max
local max: (1.633,17.419) as ( x, A )

60. 32) Maximize the area
(x,y) of the rectangle
A = (2x)(y)
y y
2
x x
y= 8− x
2
2 A = (2x)(8 − x )
y= 8− x
graph this cubic and look for a local max
local max: (1.633,17.419) as ( x, A )
max area is 17.419 sq. units
dimensions are 3.266 x 5.333 units

61. HW #8
“All you need is a plan, a road map, and the
courage to press on to your destination.”
Earl Nightingale