Volumes of Cones

Diana Herwono D 0861 006

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 θ.

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:

2π × rbase =  rθ

rbase = rθ / (2π).

The area of the base can therefore be calculated:

Abase = π × rbase 2 = π × ( rθ / (2π))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 + (rθ/2π)2

h2 = r2 - (rθ/2π)2

h = √ [r2 - (r2θ2/4π2)]

Therefore, substituting the values for rbase and h, we can find the volume of the cone.

V = 1/3 x height x base area

V = 1/3 × √ [r2 - (r2θ2/4π2)] × π (r2θ2/4π2)

2.        By using the substitution x = θ/2π , express the volume as a function of x.

x = θ/2π

θ = x2π

V = 1/3 × √ [r2 - (r2θ2/4π2)] × π (r2θ2/4π2)

V = 1/3 × √ [r2 - (r2(x2π)2/4π2)] × π (r2(x2π)2/4π2)

= 1/3 × √ [r2 - (r24x2π2/4π2)] ...