IB HL Math Portfolio - Volumes of Cones (Type III)

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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.

r2 = h2 + R2

r2 = h2 + (r?/2?)2

h2 = r2 - (r?/2?)2

h = ? [r2 - (r2?2/4?2)]

Therefore, substituting the values for rbase