1. The seniors plan to make a smell-proof enclosure for the freshmen using the Northern Parkway wall as one side. If they have 200 yards of fencing, what is the largest enclosure they could make? What is the domain of the function?

2. Coach Gouline plans to build a rectangular pen for Sherman, the official German Shepherd of his house. He plans to use the house as one side of the pen and use 140 feet of fencing to finish the enclosure. What is the largest possible area of Sherman’s grounds?

3. A capture the flag pitch is a large rectangle with a line dividing it in half. Suppose you have enough paint for 600 feet of lines. What is the largest pitch (by area) you could create?

4. John plans to dive off a 10-foot tall diving board. He jumps upwards at 20 feet per second. His height as a function of time is given by: h(t)=-16t 2 +20t+10. Find his maximum height and the amount of time he is in the air.

5. Jack grows corn on his farm. He currently has 500 bushels of corn ready to ship, but he will be able to harvest 15 more bushels for each day that he waits. Right now, corn sells for $82 per bushel, but the price is expected to drop $2 a day. How long should Jack wait to maximize his revenue? What is the domain of the function?

6. The Organization of the Petroleum Exporting Countries (OPEC) controls nearly 80% of the world’s oil reserves, and largely dictates the price of oil. Currently, OPEC produces approximately 29 million barrels of oil a day and sells each barrel for around $45 dollars per barrel. OPEC estimates that for every million additional barrels they produce, the cost of oil will decrease $1.25. What price of oil will maximize their revenue?

7. I have a 4 foot by 8 foot piece of sheet metal. I plan to make a box by cutting the corners off the box and folding the sides up. Write the volume of the box, V, as a function of x, the amount cut off each edge. Find the maximum volume. What is the domain of the function?

9. A cylinder is generated by rotating a rectangle with perimeter 20 around the y-axis. Find the volume of the cylinder as a function of x, the height. What is the domain of the function? Find the maximum volume.

10. A second cylinder is inscribed in a sphere of radius 10. What is the maximum volume of the cylinder? What is the domain of the function?

11. I have 100 yards of fencing, and I can either make one square pen or two square pens (that need not be identical). How can I maximize (or minimize) the enclosed area.

12. I have 300 yards of fencing, and I need to enclose two pens— one for freshmen and one for seniors. How can I maximize the total enclosed area?

13. Now, I plan to use an identical piece of sheet metal to make a box with a lid. Find V(x) and the maximum volume.