1. Section 5.4
I can use calculus to solve optimization problems.

2. 1. The sum of two nonnegative numbers is 20. Find the numbers
(a) if the sum of their squares is to be as large as possible.
Let the two numbers be represented by x and 20 – x.
makes x = 10 a minimum.
Maximum must occur at an endpoint.
0 and 20

3. 1. The sum of two nonnegative numbers is 20. Find the numbers
(b) If the product of the square of one number and the
cube of the other is to be as large as possible
Let the two numbers be represented by x and 20 – x. 12 20 + _
Max at 12, Min at 20
12 and 8

4. 1. The sum of two nonnegative numbers is 20. Find the numbers
(c) if one number plus the square root of the other is as large
as possible.
Let the two numbers be represented by x and 20 – x.
therefore a max

5. A rectangular pen is to be fenced in using two types of
fencing. Two opposite sides will use heavy duty fencing at $3/ft
while the remaining two sides will use standard fencing at $1/ft.
What are the dimensions of the rectangular plot of greatest area
that can be fenced in at a total cost of $3600? 3x 1y
Therefore max
The dimensions of a rectangular plot of greatest area are 300 x 900

6. 3. A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. With 800 m of fence at your disposal, what is the largest area you can enclose? x y
Therefore a max
The largest area you can enclose is 80000

7. 4. An open-top box with a square bottom and rectangular sides is to have a volume of 256 cubic inches. Find the dimensions that require the minimum amount of material. y x
therefore a min
8 x 8 x 4

8. _ +
Therefore max

9. Since A ‘ changes from
pos to neg at x = 0.860,
max of A occurs at
x = 0.860

10. 7. Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius 5. r
0.5h R
h – height of cylinder
r – radius of cylinder
R – Given radius of sphere
Therefore a max