Applying Volume Methods: Coconut and Pineapple By: Eve Pan, Wendy Zhang, and Yi Zhang AP Calculus BC Ms. Zhao Period 7

History “If I have seen further it is by standing on the shoulder of Giants.” – Isaac Newton The fathers of modern calculus: Isaac Newton and Gottfried Leibniz Newton understood the relationship between differentiation and integration (they are inverse of each other) Leibniz developed the standard notations of calculus used today Although Newton and Leibniz are credited as the fathers of calculus, their work is actually a compilation of many mathematicians before them. The ancient Greek mathematicians Antiphon and Eudoxus made use of the method of exhaustion to find plane area by applying inscribed and circumscribed polygons. Another Greek mathematician, Archimedes of Syracuse, used the method of exhaustion to prove the theory of integration. In the 17 th century, European mathematicians were discussing the idea of derivatives and integration. Isaac Barrow, Pierre de Fermat, René Descartes, John Wallis were among the pioneers of modern calculus. Newton and Leibniz derived their ideas from these pioneers. Through the work of many mathematicians, method to find volume of solids were derived, including the disk method and using known cross sections.

Gottfried LeibnizIsaac Newton

Purpose Yi, Eve, and Wendy are on vacation in Thailand and they come across a market where they buy a fresh coconut. They decide to divide the coconut water evenly among themselves by pouring the same amount into three identical cylindrical cups of radius 4 cm and height of 15 cm. To measure this, they found that the coconut’s endosperm (the edible part of the coconut) has a diameter of 12 cm. What is the depth of the water in each cup? A little kid who wants to make pineapple juice asks how to find the volume of his pineapple, of length 20 cm and width 12 cm, using two different methods.

Measurable Quantities/Methods Coconut endosperm: Diameter: 12 cm Cylindrical cup Radius: 4 cm Height: 15 cm Pineapple without spikes: Length: 20 cm Width: 12 cm Disk Method: Known cross-section: Note: The coconut's exocarp, mesocarp, and endocarp have been removed so only the endosperm remains, which is partly liquid (called “coconut water”) and partly solid (called "coconut meat"). We will only be considering the endosperm. Note: The leaves, top and bottom, and outer layer, including the spikes, of the pineapple have been removed so that there is edible part remains. We will only be considering the edible part.

Coconut Fact: 90% of a young coconut’s endosperm is coconut water

Problem 1: Coconut 12 C

Problem 2: Pineapple, Method 1 6 – P

Problem 2: Pineapple, Method 1

Problem 2: Pineapple, Method S 218

Citations Images: exquisite-seaside-tourist-icon-vector-material2.jpg content/uploads/2009/02/beach_stuff.jpg istockphoto.com pina-colada.php pineapple.php coconut.php Godfrey Kneller's 1689 portrait of Isaac Newton Christoph Bernhard Francke’s portrait of Gottfried Leibniz (“The History of Calculus” by Arthur Rosenthal) ewton.htm (“Isaac Newton’s Dispute with Gottfried Leibniz A Question of Priority: Who Discovered Calculus First”)

Summary Eve, Wendy, and Yi were only able to fill their cup to a 5.4 cm height. They thought it was better not to share a coconut in the future. Using two different methods, they were was able to find the volume of the pineapple for the little boy, and arrive at the same answer: cubic centimeters. But the little kid decided not to make pineapple juice, so they sliced the pineapple for a barbecue. The next week, they saw Ms. Zhao on the beach and she asked them to find … To be continued …