Unit Five (Part 1) AP Calculus Take-Home Packet Optimization
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1 Name: Date: Period: SAVE THE PUPS! One of the world's most adorable marine mammals is at risk of etinction. Your marine mammal research team has received a $1,000 grant to carry out an intervention to increase its population by this time net year. Your goal then is to maimize the population one year from now, hoping to publish a breakthrough paper and get more funding to do a bigger project to save the species. Based on your research data, your team estimates that if M gallons of medicine and N gallons of nutrients are added to their marine habitat, then by net year this will save M NM 1N lives. (The habitat is huge, so there is no need to worry about "overdose"). A gallon of medicine costs $5, and a gallon of nutrients costs $5. a) What is the cost of M gallons of medicine and N gallons of nutrients? b) To maimize, within your budget, the number of lives saved by net year, which combinations of medicine and nutrients would you consider putting into the habitat? c) Which combination would you use? How many adorable marine mammals will you have saved by this time net year? Page 1 of 8
2 JOKING AROUND WITH OPTIMIZATION PROBLEMS Directions: Solve each of the problems below. Find the correct answer in the column at the right. Then place the corresponding word in the appropriate blank to answer the two corny questions at the bottom of the page. 1. Determine the constant a in order that the function y will have a point of inflection at 1. a. Determine the volume of the largest right circular cylinder that can be inscribed in a given circular cone with radius 3 and height Determine the minimum point(s) of y. y 1 have a 4. Where does the graph of 3 point of inflection? AT,0 BE 4,0 COMPOSING 9 1 DECOMPOSING EXTRA 54 FORTUNE TELLER LARGE 1,0 MEDIUM Sketch the graph of y 4 4. MIDGET 6. Sketch a smooth curve y f illustrating f 3 4, f 0 for 3, f 3 0, f 0 for Determine the constant k in order that the f will have a relative minimum at. function k SMALL HE STILL IS 1 Question #1: What were the headlines after a midget fortuneteller escaped from jail? #6 #7 #3 #4 Question #: Why does Beethoven spend all his time erasing music? #5 #1 # Page of 8
3 Cat Food Can Problem The Carnation Company of L.A. produces and distributes Friskies Buffet Cat Foods. All flavors come in cylindrical containers. A si ounce can of Friskies Buffet Mied Grill has a volume of roughly 14.5 cubic inches and has the following dimensions: height = 1.75 inches and radius = 1.6 inches. But do the dimensions of this Friskies can have the smallest surface area that contains 14.5 cubic inches? Sounds like a calculus problem... Determine the radius and height of the can having a volume of 14.5 cubic inches and that has the smallest surface area. Why do Carnation and other companies package cat food and tuna and chicken in short cans that do not minimize the surface area? Go to to find out why!! CLASS, TAKE YOUR SEATS Can you fill in the first initial of each student in this math teacher s seating chart using only the clues below? Page 3 of 8
4 CLUES: 1. All students are located at integral coordinates in the y-plane. The -coordinates belong to the set {, 1, 0, 1, }, and the y-coordinates belong to the set { 1, 0, 1,, 3}.. Wallis is seated on the line which is normal to the curve point. f Newton is seated at a point of inflection of 3 4. Euler sits at the point on the curve f 4 at its minimum y which is nearest to Boole. f MacLaurin is located at the relative maimum point of the function 3 f Saccheri is seated at the absolute maimum point of the function 7. Riemann s seat is one of the critical points of the curve 8. The function k f f has a point of inflection at 1. Zeno sits at this point. y Boole is seated at the absolute maimum point on the curve Page 4 of 8
5 10. Archimedes is located at one of the vertices of the rectangle with the largest area that can be drawn with its upper vertices on the line y 1 and its lower vertices on the parabola 11. Thales sits at a point on the curve 3 f 6 43 where the slope is Leibniz sits at a point on the curve y cos where the 99 th derivative of that curve is 0. y. 13. Kronecker sits on the line which is tangent to the curve y 4 35 at the point (3, 5). 14. Fermat is seated at the point of inflection of the curve 3 y Descartes is located at one of the critical points of the curve y Cantor is located on the line tangent to the curve y 10 5 at its maimum point. 17. Gauss sits at the absolute maimum point on the curve [ 1, ]. 3 4y 3 7 over the interval 18. Viete s seat is collinear with those of Gauss and Kronecker. f Heron is located at the point of inflection of the curve Pascal lies on the line tangent to the curve 1y 16 6 at its point of inflection. Page 5 of 8
6 CLUE Worksheet for Class, Take Your Seats For each problem, write down all possible answers from the given domain and range. NAME CLUE Possible Ordered Pairs for the Seat 1 none Wallis Newton 3 Euler 4 MacLaurin 5 Saccheri 6 Riemann 7 Zeno 8 Boole 9 Archimedes 10 Thales 11 Leibniz 1 Kronecker 13 Fermat 14 Descartes 15 Cantor 16 Gauss 17 Viete 18 Heron 19 Pascal 0 Page 6 of 8
7 LET S GO OUT TO THE BALL GAME What was the longest home run in major league baseball?? In 1919, Babe Ruth his what some eperts called the longest home run ever recorded in major league baseball. In an ehibition game between the Boston Red So and the New York Giants, he sent the ball into a parabolic orbit. The trajectory of the bell is given by the equation y , where represents the horizontal distance in feet and y the vertical distance in feet of the ball from home plate. What was the greatest height reached by the ball? How far from home plate did the ball land? Solution #1: Enter the equation in Y1. Decide on an appropriate window. Use the Zoom feature to zero in on the maimum values. Use the Math menu to get an accurate value to answer the questions above. Solution # (Show your work for this one): Use calculus and algebra to solve the problem. Page 7 of 8
8 EXTRA CREDIT PROBLEM This is the infamous Question 6 on the 198 AP Eam, which was the focus of the movie Stand and Deliver. 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 epensive tank? Page 8 of 8
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