Begin with a circular piece of paper with a 4-in. radius as shown in
(a). Cut out a sector with an arc length of x. Join the two edges of the
remaining portion to form a cone with radius r and height h, as shown
in (b).
(a) Explain why the circumference of the base of the cone is 8𝜋 − 𝑥
(b) Express the radius r as a function of x.
(c) Express the height h as a function of x.
(d) Express the volume V of the cone as a function of x

Let's work through each part of the problem step by step.

### Given:
- The radius of the original circular piece of paper is \( R = 4 \) inches.
- A sector with an arc length of \( x \) is removed, and the remaining paper is joined to form a cone.

### (a) Why is the circumference of the base of the cone \( 8\pi - x \)?

1. The entire circumference of the original circular piece of paper is given by:
   \[
   C = 2\pi R = 2\pi \cdot 4 = 8\pi
   \]
2. When a sector with an arc length of \( x \) is cut out, the remaining arc length (which becomes the circumference of the cone's base) is:
   \[
   8\pi - x
   \]

Therefore, the circumference of the base of the cone is \( 8\pi - x \).

### (b) Express the radius \( r \) as a function of \( x \)

1. The circumference of the cone's base is \( 2\pi r \).
2. Since the circumference of the cone's base is equal to the remaining arc length after cutting, we have:
   \[
   2\pi r = 8\pi - x
   \]
3. Solving for \( r \):
   \[
   r = \frac{8\pi - x}{2\pi} = 4 - \frac{x}{2\pi}
   \]

Thus, the radius \( r \) of the cone is:
\[
r(x) = 4 - \frac{x}{2\pi}
\]

### (c) Express the height \( h \) as a function of \( x \)

1. The slant height of the cone (the distance from the tip of the cone to the edge of the base) is equal to the radius of the original circular piece, \( R = 4 \).
2. We can use the Pythagorean theorem to relate the height \( h \), radius \( r \), and slant height \( R \) in the cone:
   \[
   h = \sqrt{R^2 - r^2}
   \]
3. Substitute \( R = 4 \) and \( r = 4 - \frac{x}{2\pi} \) (from part (b)):
   \[
   h(x) = \sqrt{4^2 - \left(4 - \frac{x}{2\pi}\right)^2}
   \]
   Simplifying further:
   \[
   h(x) = \sqrt{16 - \left(16 - 8\frac{x}{2\pi} + \frac{x^2}{4\pi^2}\right)}
   \]
   \[
   h(x) = \sqrt{8\frac{x}{\pi} - \frac{x^2}{4\pi^2}}
   \]

So, the height \( h \) in terms of \( x \) is:
\[
h(x) = \sqrt{8\frac{x}{\pi} - \frac{x^2}{4\pi^2}}
\]

### (d) Express the volume \( V \) of the cone as a function of \( x \)

The volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]

Substitute \( r = 4 - \frac{x}{2\pi} \) and \( h = \sqrt{8\frac{x}{\pi} - \frac{x^2}{4\pi^2}} \):

1. First, find \( r^2 \):
   \[
   r^2 = \left(4 - \frac{x}{2\pi}\right)^2 = 16 - 8\frac{x}{\pi} + \frac{x^2}{4\pi^2}
   \]
2. Now, substitute \( r^2 \) and \( h \) into the volume formula:
   \[
   V(x) = \frac{1}{3} \pi \left(16 - 8\frac{x}{\pi} + \frac{x^2}{4\pi^2}\right) \sqrt{8\frac{x}{\pi} - \frac{x^2}{4\pi^2}}
   \]

This is the volume \( V \) as a function of \( x \). It can be simplified further, but this form accurately represents the volume of the cone.

Would you like a further breakdown of any of these steps or simplifications? 

---

Here are some related questions to explore further:

1. What happens to the cone's volume as \( x \) approaches 0?
2. How does the height \( h \) change as \( x \) increases?
3. Could you solve for the maximum possible volume of the cone given a range for \( x \)?
4. How does the shape of the cone (height and radius) change as \( x \) varies?
5. How would the problem change if the initial radius of the paper were different?

**Tip:** Always double-check the physical dimensions of derived formulas to ensure they make sense in practical applications!

Begin with a circular piece of paper with a 4-in. radius as shown in (a). Cut out a sector with an arc length of x. Join the two edges of the remaining portion to form a cone with radius r and height h, as shown in (b). (a) Explain why the circumference of the base of the cone is 8π − x (b) Express the radius r as a function of x. (c) Express the height h as a function of x. (d) Express the volume V of the cone as a function of x.

Learn how to calculate the radius, height, and volume of a cone formed by cutting a sector from a circular piece of paper with a radius of 4 inches. This problem covers key mathematical concepts including the volume of a cone, the Pythagorean theorem, and the relationship between arc length and the circumference of a circle. Suitable for students in Grades 9-12.

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