Question

A wire of length \(x\) is bent into the shape of a circle. (a) Express the circumference \(C\) of the circle as a function of \(x\). (b) Express the area \(A\) of the circle as a function of \(x\).

Short answer

a) \( C(x) = x \) b) \( A(x) = \frac{x^2}{4\pi} \)

Step-by-step solution

Title - Understanding the Given Wire and Circle

Given a wire of length \(x\), the whole wire is bent into the shape of a circle. Therefore, the length of the wire represents the circumference \(C\) of the circle.

- Expressing the Circumference as a Function of Length

Since the circumference of a circle is represented by \(C = 2\pi r\), and given that the length of the wire is \(x\), we can set these equal: \[ C = x \] Therefore, the circumference as a function of \(x\) is \[ C(x) = x \]

- Finding the Radius in Terms of Length

To find the area, first determine the radius \(r\) in terms of the length of the wire, using the earlier relationship \(C = 2\pi r\). Solving for \(r\): \[ r = \frac{x}{2\pi} \]

- Expressing the Area as a Function of Length

The area \(A\) of the circle is calculated using the formula \(A = \pi r^2\). Substitute \(r\) from the previous step: \[ A = \pi \left(\frac{x}{2\pi}\right)^2 \] Simplify the expression: \[ A = \pi \left(\frac{x^2}{4\pi^2}\right) = \frac{x^2}{4\pi} \] Therefore, the area as a function of \(x\) is \[ A(x) = \frac{x^2}{4\pi} \]

Key concepts

circumference function

To begin with, let's understand what the circumference function is. The circumference of a circle is the total distance around it. When you bend a wire of length \(x\) into a circular shape, this length becomes the circle’s circumference. Mathematically, you can express the circumference \(C\) of a circle using the radius \(r\). The formula is: \[ C = 2\pi r \] Given that the length of the wire \(x\) equals the circumference, we can say: \[ C = x \] This shows the circumference is directly equal to the length of the wire. Therefore, if you know the length of the wire, you automatically know the circumference of the circle.

area function

Next, let’s talk about the area function. The area of a circle is the region enclosed by its circumference. It’s calculated using the radius. The formula for the area \(A\) is: \[ A = \pi r^2 \] We already know the circumference \[ C = x = 2\pi r \], so we can solve for the radius \(r\): \[ r = \frac{x}{2\pi} \] Now, substitute this value of \(r\) back into the area formula: \[ A = \pi \left(\frac{x}{2\pi}\right)^2 \] Simplifying this expression gives: \[ A = \pi \cdot \frac{x^2}{4\pi^2} = \frac{x^2}{4\pi} \] This final formula shows the area \(A\) as a function of the length of the wire \(x\).

radius calculation

Lastly, calculating the radius is essential for finding both the circumference and the area. In this exercise, we determined the radius \(r\) by using the circumference formula. Knowing the wire length \(x\), equate it to the circumference: \[ C = x = 2\pi r \] Solving for \(r\) gives: \[ r = \frac{x}{2\pi} \] This calculated radius is crucial, as it helps transform circumference and area functions. Using the radius value, you can easily find how much space the circle occupies or how long the boundary is. To summarize:

• The circumference is directly equal to the wire length;
• The area depends on the square of the wire length divided by \(4\pi\);
• The radius is found by dividing the wire length by \(2\pi\).

These relationships simplify solving related circle geometry problems.