A conical cup is made from a circular piece of paper with radius 6 cm by cutting out a sector and joining the edges as shown below.

a) (4 + 4 + 2 marks) The formula for the volume of a cone of height h and radius of base r, as shown above, is
V=πr2h3.
Find V(θ) by finding r(θ) and h(θ) first.

To solve the problem, we'll begin by understanding how the conical cup is formed and how the radius \( r \) and height \( h \) of the cone depend on the angle \( \theta \) of the sector cut out from the circular piece of paper.

### Step 1: Determine the Radius \( r(\theta) \) of the Cone's Base
Given:
- The original radius of the circular piece of paper is \( R = 6 \) cm.
- When a sector with central angle \( \theta \) is cut out, the remaining arc of the circle becomes the circumference of the base of the cone.

The circumference \( C \) of the base of the cone is given by:
\[
C = 2\pi r
\]
This is equal to the length of the remaining arc:
\[
C = (2\pi - \theta)R
\]
Thus, we have:
\[
2\pi r = (2\pi - \theta) \cdot 6
\]
\[
r(\theta) = \frac{(2\pi - \theta) \cdot 6}{2\pi} = 6 - \frac{3\theta}{\pi}
\]

### Step 2: Determine the Height \( h(\theta) \) of the Cone
The height \( h \) can be determined using the Pythagorean theorem. The slant height \( l \) of the cone is the radius of the original circle:
\[
l = R = 6 \text{ cm}
\]
The relationship between the height \( h \), radius \( r \), and slant height \( l \) is:
\[
h = \sqrt{l^2 - r^2}
\]
Substituting \( r(\theta) \) from Step 1:
\[
h(\theta) = \sqrt{6^2 - \left(6 - \frac{3\theta}{\pi}\right)^2}
\]
\[
h(\theta) = \sqrt{36 - \left(36 - 36\cdot\frac{\theta}{\pi} + \frac{9\theta^2}{\pi^2}\right)}
\]
\[
h(\theta) = \sqrt{36\left(\frac{\theta}{\pi} - \frac{\theta^2}{\pi^2}\right)}
\]

### Step 3: Determine the Volume \( V(\theta) \) of the Cone
The volume \( V \) of a cone is given by:
\[
V = \frac{1}{3}\pi r^2 h
\]
Substituting \( r(\theta) \) and \( h(\theta) \):
\[
V(\theta) = \frac{1}{3}\pi \left(6 - \frac{3\theta}{\pi}\right)^2 \cdot \sqrt{36\left(\frac{\theta}{\pi} - \frac{\theta^

Learn how to calculate the volume of a conical cup formed from a circular piece of paper. This problem involves using geometry, trigonometry, and the Pythagorean theorem to find the volume based on the angle of the sector cut. Suitable for Grades 10-12.

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