Volumes of Cones
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IB Mathematics HL Portfolio (Type III) Volumes of Cones By Diana Herwono IB Candidate No: D 0861 006 May 2003 In this assignment, I will use WinPlot, a graphing display program. 1. Find an expression for the volume of the cone in terms of r and ?. The formula for the volume of a cone is: V = 1/3 x height x base area To find the base area, we must find the radius of the base. The circumference of the base of the cone = the length of arc ABC Therefore: 2? ? rbase = r? rbase = r? / (2?). The area of the base can therefore be calculated: Abase = ? ? rbase 2 = ? ? ( r? / (2?))2 Next, we must find the height of the cone, h. Notice that for the cone, h rbase r is a right angled triangle, with r as the hypotenuse. Therefore, using the Pythagorean Theorem, we can find h.
The value of r will change the volume of the cone, but x will remain the same). Therefore we can replace the r in the function with any constant number. And in this project, 1 is used for replacing r since it makes the function easier to calculate. Thus: Vcone = 1/3 ? ? [r2 - (r2x2)] ? ? (r2x2) Vcone = 1/3 ? ?[12 - (x212] ? ? x212 Vcone = 1/3 ? ?(1 - x2) ? ? x2 In this function, 1/3 ? is constant. The shape of the function will be x2 ? ?(1 - x2) To find the maximum value of the cone, we can either: Graph Vcone = x2 ? ?(1 - x2) and find the maximum value or Graph the derivative of Vcone and find the zero/root of the function Graph of Vcone (grey) and its derivative (orange) The two maximum values are at x = ?0.8174. They both give a maximum cone volume of 0.3871.
? (1-x)2 r2 Finding the sum of the two cones: V = Vcone (A) + Vcone (B) = 1/3 ? ?(r2 - x2r2) ? ? x2r2 + 1/3 ? ?[r2 - (1-x)2 r2] ? ? (1-x)2 r2 Like Question 3, we can replace the constant r with 1 for easier calculation. We get: V = 1/3 ?(1 - x2) ? ? x2 + 1/3 ? ?[1 - (1-x)2 ] ? ? (1-x)2 Graphing this equation, we can find the maximum sum of the volumes of the two cones. Graph of Vcone (A) (grey), Vcone (B) (red), and Vsum (blue). The two maximum cone volumes we get are both 0.4566. The x values that give this maximum cone volume are 0.6760 and 0.3240. Notice that these two x values are just complements of each other (they add up to 1). This means that x = 0.3240 and x - 1 = 0.6760. The maximum volume = 0.6760. Similarly, this answer can also been found by finding the derivative of Vsum and equating it to zero. The zero/root of the function will be the maximum value. ?? ?? ?? ?? Diana Herwono D 0861 006 1
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