1. Using Calculus to Solve Optimization Problems
Section 5.4
Using Calculus to Solve Optimization Problems
5.3
Pick up packet out of folder

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. 5. Find the largest possible value of 2x + y if x and y are the lengths of the sides of a right triangle whose hypotenuse is
units long. x y 2 + _
Therefore x = 2 is a max

9. 6. A right triangle of hypotenuse 5 is rotated about one of its legs to generate a right circular cone. Find the cone of greatest volume. x y 5
Therefore max

10. Determine the area of the largest rectangle that may be
Inscribed under the curve 1 _ +
Therefore max

11. Since f’ changes from neg to pos, we have a minimum
8. (calculator required) A poster is to contain 100 square inches of picture surrounded by a 4 inch margin at the top and bottom and a 2 inch margin on each side. Find the overall dimensions that will minimize the total area of the poster. 4 2 x y
Since f’ changes from neg to pos, we have a minimum

12. 9/2 + _
Therefore min

13. + _
Therefore max

14. Since f’ changes from pos to neg, we have a maximum
(calculator required) Find the dimensions of the rectangle
with maximum area that can be inscribed in a circle of radius 10.
Since f’ changes from pos to neg, we have a maximum

15. Max at x = 7.46 since f ‘ changes from pos to neg
At 7.46, r = , c = , or P =
Max profit is $163,000 which occurs when 7460 units are made

16. A tank with a rectangular sides is to be open at the top. It is to be
constructed so that its width is 4 m and its volume is 36 cubic meters. If
building the tank costs $10 per square meter for the base and $5 per square
meter for the sides, what is the cost of the least expensive tank? y x 4

17. CALCULATOR REQUIRED
Minimum since f ‘ (x) changes from
neg to pos at –0.426

18. Since A ‘ changes from neg to pos, min area at t = 2

19. + _
Therefore max

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

21. Consider the set of all right circular cylinders for which the
sum of the height and diameter is 18 inches. What is the radius
of the cylinder with the maximum volume? X

23. _ +

24. Two possibilities where
CALCULATOR REQUIRED
Two possibilities where
D ‘ changes from neg
to pos, denoting a min

25. 22. 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