4. Use the Strategy
• What is the quantity to be optimized?
– The volume
• What are the measurements (in terms of
x)?
• What is the variable which will
manipulated to determine the optimum
volume?
60”
• Now use calculus
x
principles
30”
7. Optimization
Maximizing or minimizing a quantity based on a given situation
Requires two equations:
Primary Equation
what is being maximized or minimized
Secondary Equation
gives a relationship between variables
8. To find the maximum (or minimum) value of a function:
1
Write it in terms of one variable.
2
Find the first derivative and set it equal to zero.
3
Check the end points if necessary.
9. Ex. 1 A manufacturer wants to design an open box
having a square base and a surface area of 108 in2.
What dimensions will produce a box with maximum
volume?
Since the box has a square
h base, its volume is
V = x 2h
x
x
Note: We call this the primary
equation because it gives a
formula for the quantity we
wish to optimize.
The surface area = the area of the base + the area of the 4 sides.
S.A. = x2 + 4xh = 108
We want to maximize the volume,
so express it as a function of just
one variable. To do this, solve
x2 + 4xh = 108 for h.
10. h
V
108 x 2
4x
2
xh
Substitute this into the Volume equation.
108 x
x
4x
2
2
27 x
x3
4
To maximize V we find the derivative and it’s C.N.’s.
dV
dx
27
3x 2
4
0
3x2 = 108
C.N.' s x
We can conclude that V is a maximum when x = 6 and
the dimensions of the box are 6 in. x 6 in. x 3 in.
6
12. x 2 that is closest to (0,3).
Find the point on f x
2.
Minimize distance
Secondary
Primary
d
x 0
d
x
2
2
y 3
y 3
y x2
2
2
***The value of the root will be smallest
when what is inside the root is smallest.
d
d x
d x
x
2
y 3
2
2
2
Intervals:
2
x
x 3
x 2 x 4 6x 2 9
d x x 4 5x 2 9
d ' x 4x3 10x
4x 3 10x 0
2x 2x 2 5 0
x 0
2x 2 5 0
x 0
x
5
2
Test values:
5
2
,
5
2
,0
0,
5
2
5
2
,
3
1
1
3
dec
inc
dec
inc
d ’(test pt)
d(x)
rel max
rel min
x
5
2
5 5
2 2
,
rel min
x
5
2
5 5
2 2
,
13. 2.
A rectangular page is to contain 24 square inches of print. The
margins at the top and bottom are 1.5 inches. The margins on each side are
1 inch. What should the dimensions of the print be to use the least paper?
14. 2.
A rectangular page is to contain 24 square inches of print. The
margins at the top and bottom are 1.5 inches. The margins on each side are
1 inch. What should the dimensions of the print be to use the least paper?
Primary
A
A( x )
Secondary
xy 24
x 2 y 3
x 2
24
x
48
x
3x 48 x
1
A '( x ) 3
y
3
24 3 x
1.5
6
30
48
x2
y
y
2
24
x
24 in
1
24
4
y 3
y
1
x
1.5
6
x 2
Smallest
Largest
(x is near zero)
x 0
crit #'s: x 0, 4
(y is near zero)
x 24
Page dimensions: 9 in x 6 in
0,4
4,24
Test values:
Print dimensions: 6 in x 4 in
Intervals:
1
10
dec
inc
A ’(test pt)
A(x)
rel min
x 4
16. 1.
Find two positive numbers whose sum is 36 and
whose product is a maximum.
Primary
P
P x
Secondary
xy
x 36 x
y
P '( x ) 36 2x
36 2x 0
x 18
Intervals:
Test values:
0,18
1
18,36
inc
dec
20
P ’(test pt)
P(x)
rel max
x 18
x y 36
y 36 x
36 18 18
18,18
17. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden
along the side of a barn. What is the maximum area that
you can enclose?
A x 40 2x
x
x
A 40 x 2 x 2
A
40 2x
w
l
x
40 2x
0 40 4x
w 10 ft
l
40 4x
20 ft
4x 40
x 10
There must be a
local maximum
here, since the
endpoints are
minimums.
18. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden
along the side of a barn. What is the maximum area that
you can enclose?
A x 40 2x
x
x
A 40 x 2 x 2
A
40 2x
w
l
x
40 2x
A 10 40 2 10
0 40 4x
w 10 ft
l
40 4x
20 ft
4x 40
x 10
A 10 20
A 200 ft 2
19. Example 5:
What dimensions for a one liter cylindrical can will
use the least amount of material?
We can minimize the material by minimizing the area.
We need another
equation that
relates r and h:
V
r 2h
3
1 L 1000 cm
1000
1000
r2
r 2h
A 2 r 2 2 rh
area of
ends
A 2 r
lateral
area
1000
2 r
r2
2
A 2 r
2
h
A
4 r
2000
r
2000
r2
20. Example 5:
What dimensions for a one liter cylindrical can will
use the least amount of material?
r 2h
V
A 2 r 2 2 rh
3
1 L 1000 cm
1000
1000
r2
r 2h
lateral
area
1000
2 r
r2
A 2 r2
h
A 2 r
1000
5.42
area of
ends
2
h
h 10.83 cm
A
0
2
4 r
4 r
2000
r2
4 r
2000 4 r 3
500
2000
r
2000
r2
2000
r2
r3
500
r
3
r
5.42 cm
21. Notes:
If the function that you want to optimize has more than
one variable, use substitution to rewrite the function.
If you are not sure that the extreme you’ve found is a
maximum or a minimum, you have to check.
If the end points could be the maximum or minimum,
you have to check.
22. Example #1
• A company needs to construct a cylindrical
container that will hold 100cm3. The cost
for the top and bottom of the can is 3 times
the cost for the sides. What dimensions are
necessary to minimize the cost.
2
r
V
h
r h
SA 2 rh 2 r
2
23. Minimizing Cost
V
r 2h
100
SA 2 rh 2 r 2
r
SA 2 r
SA
r 2h
100
100
r
2
2 r
200
2 r2
r
Domain: r>0
2
C (r )
C (r )
h
2
200
6 r2
r
200
r
2
12 r
24. Minimizing Cost
C (r )
0
200
r2
200
r
200
r2
2
3
r
C (1.744 ) 0
12
Concave up – Relative min
------ +++++
12 r
1.744
0
C' changes from neg. to pos.
200 12 r 3
3
200
12
r
12 r
12 r
100
C (r )
r
3 50
3
1.744
100
r
2
Rel. min
h
h 10.464
The container will have a radius of
1.744 cm and a height of 10.464 cm
