AP Calculus AB (Vahsen) Name: ________________________________ Optimization Project (PART 1) Part 1 of this project (this paper) is due at the end of class on December 15/16. Part 2 of this project will be completed in class on December 15/16. Directions for Part 1: You must complete FIVE of the following optimization problems. These problems will be worth half of your project grade (50 points – 10 points each). Numerical answers to these problems will be posted online and in rooms 224 and 218. Make sure to include units (where appropriate) and to JUSTIFY YOUR ANSWER. You may complete the bonus problem for 5 points of extra credit (it will not count as one of your graded problems). 1. Find the point on the graph of the function f x x that is closest to the point (4, 0). 2. A rancher has 200 feet of fencing with which to enclose to adjacent rectangular corrals. What dimensions should be used to that the enclosed area will be a maximum? AP Calculus AB (Vahsen) Name: ________________________________ 3. A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank costs $10 per square meter for the base and $5 per square meter for the sides, what is the cost of the least expensive tank? 4. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet. 5. A rectangle is bounded by the x-axis and the semicircle y 9 x 2 . What length and width should the rectangle have to that its area is a maximum? AP Calculus AB (Vahsen) Name: ________________________________ 6. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius r. 7. Determine the dimensions of a rectangular solid (with a square base) with a maximum volume if its surface area is 337.5 square centimeters. 8. A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. AP Calculus AB (Vahsen) Name: ________________________________ 9. Find the points on the hyperbola y2 – x2 = 4 that are closest to the point (2, 0). 10. A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a square cross section) of 108 inches. Find the dimensions of the package of a maximum volume that can be sent. BONUS (DO ON SEPARATE PAPER): A pipe is being carried (so that it is parallel to the floor) around a corner from a hallway that is 8 feet wide to a hallway that is 4 feet wide. What is the maximum length of the pipe that can be carried around the corner without getting stuck?