1. Optimization Problems
1. Express the number 10 as a sum of two nonnegative numbers whose product is as large as possible.
x = 1st number
y = 2nd number
x + y = 10
P = x y 5
The number 5 and 5 will make the product a maximum because
the slope goes from positive to negative when x = 5

2. Optimization Problems
2. A rectangular field is bounded by a fence on 3 sides and a straight stream on the fourth side. Find the dimensions of the field with the maximum area that can be enclosed with 1000 feet of fence.
x = width
y = length x x
2x + y = 1000
A = x y x
Since the second derivative is negative at a critical point,
250 ft x 500 ft will be the maximum area.

3. Optimization Problems
3. Find the dimensions of the biggest rectangle that can be inscribed in the right triangle with dimensions of 6 cm, 8 cm, and 10 cm.
x = width
y = length
8 cm
6 cm
10 cm
8 - y x y y
A = x y x
Since the second derivative is negative at a
critical point, 6 cm2 will be the maximum area.

4. Optimization Problems
4. A piece of wire of 12 cm can be bent into a circle, a square, or cut into two pieces to make both a square and a circle. How should the wire be cut to maximize and minimize the area?
x x

5. Optimization Problems
4. A piece of wire of 12 cm can be bent into a circle, a square, or cut into two pieces to make both a square and a circle. How should the wire be cut to maximize and minimize the area? x A
x x
9 cm2
5.04 cm2
12 cm
11.46 cm2

6. Optimization Problems
5. Find the length of the sides of an isosceles triangle with perimeter of 12 cm whose area is a maximum.
h x
6 – x
x x h
12 – 2x

7. Optimization Problems
5. Find the length of the sides of an isosceles triangle with perimeter of 12 cm whose area is a maximum.
h x
6 – x
x x h
12 – 2x 4
x = 4 is a maximum by the 1st derivative test. The sides are 4 cm x 4 cm x 4 cm

8. Optimization Problems
6. A triangle is inscribed in a semicircle of radius 10 cm so that one side is along the diameter. Find the dimension of the triangle with the maximum area.
b h
20 cm

9. Optimization Problems
6. A triangle is inscribed in a semicircle of radius 10 cm so that one side is along the diameter. Find the dimension of the triangle with the maximum area.
b h
20 cm
h is a maximum by the 1st derivative test.

10. Optimization Problems
7. A cattle rancher allows 20 bulls per acre. On the average, the bulls weigh 2000 lb. Estimated show that the average market per bull will decrease by 50 lbs for each additional bull. How many bulls per acre should be allowed in order maximize the total market weight?
x is a maximum by the 2nd derivative test so the farmer should allow 30 bulls per acre.

11. Optimization Problems
8. How should two non-negative numbers be chosen that their sum is 1 and the sum of their squares is (a) as large as possible, and (b) as small as possible?
x sum
1/2 1/2
0, 1 is the maximum sum
½ , ½ is the minimum sum