Question

how long is the arc intersected by a central angle of \\( \frac{\pi}{2} \\) radians in a circle with a radius of 4.5 cm? round your answer to the nearest tenth. use 3.14 for \\( \pi \\).
0.3 cm
0.7 cm
2.9 cm
7.1 cm

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) when the central angle \( \theta \) is in radians is \( s = r\theta \), where \( r \) is the radius of the circle.

Step2: Substitute the given values

We are given that \( r = 4.5 \) cm and \( \theta=\frac{\pi}{2} \) radians. We use \( \pi = 3.14 \), so first calculate \( \theta=\frac{3.14}{2}=1.57 \) radians. Then substitute into the formula: \( s=4.5\times1.57 \).

Step3: Calculate the product

\( 4.5\times1.57 = 7.065 \). Rounding to the nearest tenth, we look at the hundredth place (6), which is greater than 5, so we round up the tenth place. \( 7.065\approx7.1 \) cm. Wait, no, wait, I made a mistake. Wait, \( \frac{\pi}{2} \) radians, \( r = 4.5 \). Wait, no, wait, the formula is \( s = r\theta \), so \( \theta=\frac{\pi}{2} \), so \( s = 4.5\times\frac{\pi}{2} \). Let's recalculate. \( \pi = 3.14 \), so \( \frac{3.14}{2}=1.57 \), then \( 4.5\times1.57 = 7.065 \), but that's not matching the options? Wait, no, wait, maybe I misread the angle. Wait, the angle is \( \frac{\pi}{6} \)? Wait, no, the original problem says \( \frac{\pi}{2} \)? Wait, the options have 7.1, but let's check again. Wait, maybe the angle is \( \frac{\pi}{6} \)? Wait, no, the user's problem says \( \frac{\pi}{2} \). Wait, no, let's check the options. The options are 0.3, 0.7, 2.9, 7.1. Wait, 4.5 times \( \frac{\pi}{2} \) is \( \frac{4.5\times3.14}{2}= \frac{14.13}{2}=7.065\approx7.1 \), which is option D. But wait, maybe the angle is \( \frac{\pi}{6} \)? Wait, no, the user's problem says \( \frac{\pi}{2} \). Wait, but let's check the calculation again. Wait, \( s = r\theta \), \( r = 4.5 \), \( \theta=\frac{\pi}{2} \). So \( s = 4.5\times\frac{\pi}{2} \). \( \pi = 3.14 \), so \( 4.5\times1.57 = 7.065 \), which rounds to 7.1. So the answer should be 7.1 cm.

Wait, but the options have 7.1 as an option. So that's correct.

Answer:

7.1 cm (corresponding to the option with 7.1 cm, e.g., if the options are labeled, say D. 7.1 cm, then D. 7.1 cm)