How long is the arc intersected by a central angle of [tex]$\frac{\pi}{2}$[/tex] radians in a circle with a radius of 4.5 cm?

Round your answer to the nearest tenth. Use 3.14 for [tex]$\pi$[/tex].

A. 0.3 cm
B. 0.7 cm
C. 2.9 cm
D. 7.1 cm



Answer :

To determine the length of the arc intersected by a central angle of [tex]\(\frac{\pi}{2}\)[/tex] radians in a circle with a radius of 4.5 cm, follow these steps:

1. Identify the given values:
- Radius ([tex]\(r\)[/tex]) of the circle: 4.5 cm
- Central angle ([tex]\(\theta\)[/tex]) in radians: [tex]\(\frac{\pi}{2}\)[/tex]

2. Use the formula for arc length:
The arc length ([tex]\(L\)[/tex]) of a circle is given by the formula:
[tex]\[ L = r \times \theta \][/tex]

3. Substitute the given values into the formula:
[tex]\[ L = 4.5 \, \text{cm} \times \frac{\pi}{2} \][/tex]

4. Use the approximation [tex]\(\pi \approx 3.14\)[/tex]:
[tex]\[ \frac{\pi}{2} \approx \frac{3.14}{2} = 1.57 \][/tex]
Thus,
[tex]\[ L = 4.5 \, \text{cm} \times 1.57 \][/tex]

5. Calculate the arc length:
[tex]\[ L = 4.5 \times 1.57 = 7.065 \, \text{cm} \][/tex]

6. Round the arc length to the nearest tenth:
[tex]\[ L \approx 7.1 \, \text{cm} \][/tex]

The length of the arc is approximately 7.1 cm.