A truck moves along a circular path and describes 100 m when it has traced out 36° at the centre. The radius of the circle is equal to
Understanding the Problem: Truck on a Circular Path
This problem involves a truck moving along a circular path. We are given the distance the truck travels along the path (which is an arc length) and the angle traced out at the center of the circle by this movement. We need to find the radius of this circular path.
The relationship between the arc length, the radius of the circle, and the central angle is a fundamental concept in geometry and trigonometry.
Key Concept: Arc Length Formula
The arc length ($\(s\)$) of a circle is the distance along the curved line. It is directly proportional to the radius ($\(r\)$) of the circle and the central angle ($\(\theta\)$) it subtends. The formula relating these quantities is:
\[s = r\theta\]
Important Note: In this formula, the angle $\(\theta\)$ must be measured in radians, not degrees.
Step-by-Step Calculation
We are given the following information:
- Arc length, \(s = 100\) m
- Central angle, \(\theta = 36^\circ\)
Our goal is to find the radius, \(r\).
Step 1: Convert the Central Angle from Degrees to Radians
The given angle is in degrees (\(36^\circ\)). We need to convert it to radians to use the arc length formula. The conversion factor is \(\pi\) radians = \(180^\circ\).
\[\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180^\circ}\]
Substituting the given angle:
\[\theta = 36^\circ \times \frac{\pi}{180^\circ} \text{ radians}\]
Simplify the fraction:
\[\theta = \frac{36}{180}\pi \text{ radians}\]
Since \(180 \div 36 = 5\), the fraction simplifies to \(\frac{1}{5}\).
\[\theta = \frac{\pi}{5} \text{ radians}\]
Step 2: Apply the Arc Length Formula
Now that we have the angle in radians and the arc length, we can use the formula \(s = r\theta\) to solve for the radius \(r\).
Substitute the known values:
\[100 \text{ m} = r \times \frac{\pi}{5} \text{ radians}\]
Step 3: Solve for the Radius \(r\)
To find \(r\), we need to isolate it in the equation. Divide both sides by \(\frac{\pi}{5}\):
\[r = \frac{100}{\frac{\pi}{5}} \text{ m}\]
To divide by a fraction, multiply by its reciprocal:
\[r = 100 \times \frac{5}{\pi} \text{ m}\]
\[r = \frac{500}{\pi} \text{ m}\]
Conclusion: Radius of the Circular Path
The radius of the circular path along which the truck moves is \(\frac{500}{\pi}\) meters.
Revision Table: Circular Path Calculation
| Quantity | Value Given | Unit | Notes |
|---|---|---|---|
| Arc Length (\(s\)) | \(100\) | meters | Distance traveled on the circle's edge |
| Central Angle (\(\theta\)) | \(36^\circ\) | degrees | Angle subtended by the arc at the center |
| Central Angle (\(\theta\)) | \(\frac{\pi}{5}\) | radians | Converted for use in formula |
| Radius (\(r\)) | \(\frac{500}{\pi}\) | meters | Calculated value |
Additional Information: Radians vs. Degrees
It is crucial to understand the difference between degrees and radians when working with angles in formulas involving arc lengths, areas of sectors, or angular velocity. Degrees are a historical unit of angle measurement, where a full circle is divided into \(360^\circ\). Radians, on the other hand, are a more natural unit in many mathematical and physical contexts, defined based on the radius of the circle. One radian is the angle subtended at the center of a circle by an arc that has a length equal to the radius of the circle. A full circle is \(2\pi\) radians, which is equivalent to \(360^\circ\). Therefore, the conversion is \(\pi\) radians = \(180^\circ\).
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