1. 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

2. 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

3. 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

4. 2. 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?
Therefore a max

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. 2 + _
Therefore x = 2 is a max

6. 4. A right triangle of given hypotenuse is rotated about one of its legs to generate a right circular cone. Find the cone of greatest volume. x y h

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

8. 6. 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.
therefore a minimum

9. 7. 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.
therefore a min
8 x 8 x 4

10. Now let L = D^2. We can do this since the minimum of D^2
8. Find the point on the curve that is a minimum distance from the point (4, 0).
Now let L = D^2. We can do this since the minimum of D^2
would be the same as the minimum of L.
Therefore a minimum

11. Since 163.14 > -3.14, max profit is $163,000 which occurs when
9. Suppose that the revenue of a company can be represented with the function r(x) = 48x, and the company’s cost function is , where x represents thousands of units and revenue and cost are represented in thousands of dollars. What production level maximizes profit, and what is the maximum profit to the nearest thousand dollars?
At 7.46, r = , c = , or P =
At 0.54, r = 25.92, c = or P = -3.14
Since > -3.14, max profit is $163,000 which occurs when
7460 units are made