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 17th 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 Leibniz Isaac 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.
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. 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:
Coconut Fact: 90% of a young coconut’s endosperm is coconut water
12 C Problem 1: Coconut 12 Let C be the region in the first quadrant bounded by the graph of , the -axis, and the -axis, as shown in the figure above. Volume of endosperm generated when C is revolved about the x-axis: or 904.779
Since 90% of the coconut endosperm consists of coconut water, the volume of coconut water,, of the coconut is: Volume of coconut water that each person is going to get: How high will the water fill the cylindrical cup? The coconut water will fill each cup with a height of 5.4 cm.
6 • P Problem 2: Pineapple, Method 1 2 18 20 – 6 Let P be the region in the first and fourth quadrant bounded by the graph of , , and the vertical lines and , as shown in the figure above. The region P is the base of the solid. For this solid, each cross section perpendicular to the -axis is a semicircle.
Problem 2: Pineapple, Method 1 Volume of the solid by using cross sections (which is half the volume of the pineapple): ◄ Finding ◄ Substitute to find volume Volume of the pineapple: The volume of the pineapple by using this method is 1423.518 .
6 Problem 2: Pineapple, Method 2 S 2 18 20 Let S be the region in the first quadrant bounded by the graph of , and the vertical lines and , as shown in the figure above. Volume of pineapple by applying disk method: The volume of the pineapple found by using disk method is 1423.518
Citations Images: http://www.resourcesgraphics.com/images/Subject-exquisite-seaside-tourist-icon-vector-material2.jpg http://www.vectorjungle.com/wp-content/uploads/2009/02/beach_stuff.jpg istockphoto.com http://www.istockphoto.com/stock-illustration-9588897-coconut-pina-colada.php http://www.istockphoto.com/stock-illustration-9488560-pineapple.php http://www.istockphoto.com/stock-illustration-9136809-coconut.php Godfrey Kneller's 1689 portrait of Isaac Newton Christoph Bernhard Francke’s portrait of Gottfried Leibniz http://www.jstor.org/pss/2308368 (“The History of Calculus” by Arthur Rosenthal) http://courses.science.fau.edu/~rjordan/phy1931/NEWTON/newton.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: 1423.518 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 …