To solve the problem, we need to use the relationship between the arc length, radius, and central angle of a circle. This relationship is given by the formula:

$s = r \theta$

where:

• $s$ is the arc length,
• $r$ is the radius,
• $\theta$ is the central angle in radians.

Step 1: Use the given data

From the problem:

• $s = 5\pi \, \text{m}$,
• $r = 4 \, \text{m}$,
• The central angle is $x^\circ$, which we must convert to radians for the formula.

In radians, the angle $\theta$ (in radians) is related to $x^\circ$ by:

$\theta = \frac{x \cdot \pi}{180}$

Step 2: Substitute into the formula

$s = r \theta$

Substitute $s = 5\pi$, $r = 4$, and $\theta = \frac{x \cdot \pi}{180}$:

$5\pi = 4 \cdot \frac{x \cdot \pi}{180}$

Step 3: Simplify the equation

Cancel out $\pi$ from both sides:

$5 = 4 \cdot \frac{x}{180}$

Simplify further:

$5 = \frac{4x}{180}$

Multiply both sides by 180 to eliminate the denominator:

$5 \cdot 180 = 4x$

$900 = 4x$

Solve for $x$:

$x = \frac{900}{4} = 225$

Final Answer:

The value of $x$ is:

$x = 225^\circ$

Let me know if you need further clarification!

Related Questions:

1. What is the relationship between degrees and radians, and how does it apply here?
2. How would the solution change if the radius of the circle was different?
3. Can you derive the formula for the arc length of a sector?
4. What happens to the arc length when the central angle doubles?
5. How does this problem relate to real-world applications of circular motion?

Tip:

Always check whether the angle is in degrees or radians before applying formulas to avoid errors.