**Chapter 3.1 - Higher Order Derivatives, Velocity and Acceleration**

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**Chapter 3.2 - Minimum and Maximum on an Interval (Extreme Values)**

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**Chapter 3.3 - Optimization Problems**

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A rectangular sheet of metal, 60 cm by 50 cm, is going to be used to make a rectangular box with an open top by cutting congruent squares from the corners. Calculate the maximum possible volume of the box

Answer: approx. 12096 cm^3

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Determine the area of the largest rectangle that can be inscribed inside a semi-circle with a radius of 25 cm.

Answer: 625 cm^2

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Determine the minimum amount of material needed to make a cylindrical can with a volume of 100 cm^3.

Answer: Approx. 119.3 cm^2

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Determine the area of the largest rectangle that can be inscribed in a right triangle if the legs adjacent to the right angle oare 11 cm and 15 cm long. The two sides of the rectangle lie along the legs.

Answer: 41.25 cm^2

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A train leaves the station at 1:00 p.m. traveling due west at 80 km/h. A second train traveling south at 70 km/h reaches the station at 3:00 p.m. At what time were the two trains closest together?

Answer: Approx 1:52

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Determine the minimum distance from the point (3,8) to the line 5x - 2y + 10 = 0

Answer: 1.67 units approx.

**Chapter 3.4 - Optimization Problems in Economics and Science**

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A rectangular piece of land is to be fenced using two kinds of fencing. The north wall and south wall will have fencing that costs $20 /m and the east and west wall will have fencing that costs $15 /m. What is the greatest area that can be fenced in for $10000.

Answer: approx. 20833 m^2

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Cornelius has printed 1000 t-shirts to sell at a rock concert. He knows that if he charges $20 per shirt, he will sell all of the shirts. For every $0.50 increase in the price of a shirt, he will sell 10 fewer shirts.

a) What price will maximize his revenue?

b) What is the maximum revenue?

Answer: a) $35 b) $24500

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Tina is building a safe with a volume of 1 m^3. The safe is a square based prism with a minimum height of 40 cm and a maximum height of 200 cm. The material to make the base costs $1.50 /cm^2, the material to make the roof costs $0.50 /m^2, and the material to make the side walls costs $0.80 /cm^2. What is the minimum cost of making the safe?

Answer: approx. $51706.44

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Bernadette needs to fence in a rectangular courtyard. The north side will have a fence that costs $10/m, and the other three sides will have fencing that costs $8/m. What is the minimum cost possible if the enclosed area is to be 100 m^2?

Answer: approx. $339.41

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Clarence the apple orchard owner knows that if he plants 50 trees on one hectare of land, each tree will yield 600 apples. For each additional tree that he plants, the average yield will decrease by 8 apples per tree. Each apple can be sold for $0.15, and it costs $4 to maintain each tree.

How many trees should Clarence grow on a hectare to maximize his profit?

Answer: 61 trees

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A cylindrical silo with a hemisphere on top needs to hold 500 m^3 of corn. The material to make the hemisphere costs $100 /m^2, while the material to make the side costs $40 /m^2. What is the minimum cost?

Answer: approx. $17072.96

Chapter 3 Review

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An offshore oil well, P, is located in the ocean 7 km from the nearest point on the shore, A. A pipeline is to take oil from P to a refinery that is 18 km along the straight shoreline from A. If it costs $50000 /km to lay pipe underwater and $35000 /km to lay pipe underground, what is the cheapest possible cost to run pipe from the oil well to the refinery?

Answer: Approx $879 950.00

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