Optimization Problems

1. Optimization Problems Applied Minimum & Maximum Problems Section 3.7

2. “Geometric Problems” Area / Perimeter / Volume / Surface Area

3. Example • A metal box (without a top) is to be constructed from a square sheet of metal 10 in. on a side by first cutting square pieces of the same size from the corners of the sheet and then folding the sides up. Find the dimensions of the box with the largest volume that can be constructed.

4. Example • A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

5. Example • What is the radius of a cylindrical soda can with volume of 512 cubic inches that will use the minimum material?

6. “Shortest Route” Usually has 2 different speeds!

7. Example • A swimmer is at a point 500 m. from the closest point on a straight shoreline. She needs to reach a cottage located 1800 m. down shore from the closest point. If she swims 4 m/sec and walks 6 m/s, how far from the cottage should she come ashore so as to arrive at the cottage in the shortest time?

8. Example • A girl standing on a straight road wants to run down the road and then diagonally through a field to her car, which she left 1 mile from the nearest point on the road. That point on the road is 2 miles from where she is now standing. If she can run 5 mph on the road, but only 3 mph through the field, at what point should she leave the road and cut through the field in order to get to her car as quickly as possible?

9. Example • It is commonly known that homing pigeons fly faster over land than over water. Assume that instead of flying 10 meters per second (as over land), they fly only 9 meters per second over water. If a pigeon is located at the edge of a river 500 meters wide and must fly to its nest 1300 meters away on the opposite edge of the river, what path would minimize its flying time?

10. “Cost or Profit”

11. Example • It costs a bus company $125 to run a bus on a certain tour, plus $15 per passenger. The capacity of the bus is 20 persons and the company charges $35 per ticket if the bus is full. For each empty seat; however, the company increases the ticket price by $2 per person. For maximum profit, how many empty seats would the company prefer? (Must use calculus).

12. Example • A bookstore can buy books from a publisher at a cost of $6/book and has been selling 200 of the books per month to the public at $30/copy. The bookstore is planning to lower its price per book to stimulate sales, and estimates that for each $2 reduction in price, 20 more books will be sold per month. At what price should the bookstore sell the books in order to generate the greatest profit?

13. Miscellaneous

14. Example • What is the largest possible product of two nonnegative numbers whose sum is 1?

15. Example • At noon a sailboat is 20 km south of a freighter. The sailboat is traveling east at 20 km/hr, and the freighter is traveling south at 40 km/hr. If visibility is 10 km, could the people on the 2 ships ever see each other?

16. Example • A plane flying due north at 750 mph flies directly over a car traveling due west at 80 mph. If the distance between the car and the plane is changing at a rate of 200 mph 1 minute later, what is the altitude of the plane?

17. Project Problem 1 • If you plan to make an open-topped box out of a sheet of tin 24” wide by 45” long by cutting congruent squares out of each corner and then bending up and soldering the resulting flaps at the corners, what should be the dimensions of the box in order to have the largest volume?

18. Project Problem 2 • A landowner wishes to use 2000 meters of fencing to enclose a rectangular region. Suppose one side of the property lies along a stream and thus doesn’t need to be fenced in. What should the lengths of the sides be in order to maximize the area?

19. Project Problem 3 • A forest ranger is in a forest 2 miles from a straight road. A car is located 5 miles down the road. If the ranger can walk 3 mi/hr in the forest and 4 mi/hr along the road, toward what point on the road should the ranger walk in order to minimize the time needed to walk to the car?

20. Project Problem 4 • An island is at a point A, 6 km offshore from the nearest point B on a straight beach. A woman on the island wishes to go to point C, 9 km down the beach from B. The woman can rent a boat for $15/km and travel by water to a point P between B and C, and then she can hire a car with a driver with a cost of $12/km and travel a straight road from P to C. Find the least expensive route for her.

21. Project Problem 5 • A cable TV co. wishes to place an amplifier station at a point on a street & run wires from the station to 3 houses. One house is adjacent to the street, & 2 are 50 ft from the street. Where should the station be located in order to minimize the total length of wire required to service all three houses? ** The distance along the street between the houses is 200 feet.

22. Project Problem 6 • The Spice-of-Life Company is preparing to create shipping crates. The co. wishes the volume of each crate to be 6 cu. Ft, w/ the crate’s base to be square b/w 1 and 2 ft on a side. Assume the material for the bottom costs $5, the sides $2, and the top $1 / sq. ft. Find the dimensions that yield the minimum cost.

23. Project Problem 7 • A plane flying 400 mph crosses at right angles 10 miles over a car going 50 mph. Two hours later how fast is the distance between the plane and the car increasing?