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Answer :
Sure! Let's solve these two questions step by step:
### Question 1
Find the radius of the circle where a central angle of 60° intercepts an arc of length 37.4 cm.
1. Understand the relationship: The length of an arc is related to the angle it subtends at the center of the circle and the radius of the circle. The formula to find the arc length [tex]\( L \)[/tex] is given by:
[tex]\[
L = r \times \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the central angle in radians.
2. Convert the angle to radians: The angle is given in degrees, so we need to convert it to radians. The conversion factor is [tex]\(\pi\)[/tex] radians = 180 degrees. For a 60° angle, the conversion is:
[tex]\[
\theta = 60^\circ \times \left(\frac{\pi}{180}\right) = \frac{\pi}{3} \text{ radians}
\][/tex]
3. Plug into the formula and solve for the radius:
- Given: [tex]\( L = 37.4 \)[/tex] cm
- Substitute the known values into the arc length formula:
[tex]\[
37.4 = r \times \frac{\pi}{3}
\][/tex]
4. Rearrange to find the radius:
[tex]\[
r = \frac{37.4}{\frac{\pi}{3}} = \frac{37.4 \times 3}{\pi}
\][/tex]
5. Calculate the radius: This gives us a radius of approximately 35.71 cm.
### Question 2
Calculate the arc length the minute hand of a watch moves in 15 minutes, where the minute hand is 1.5 cm long.
1. Understand the movement of the minute hand: The minute hand of a watch completes a full revolution (360°) in 60 minutes. Therefore, in 15 minutes, the angle it covers is:
[tex]\[
\frac{15}{60} \times 360^\circ = 90^\circ
\][/tex]
2. Convert the angle to radians:
[tex]\[
90^\circ \times \left(\frac{\pi}{180}\right) = \frac{\pi}{2} \text{ radians}
\][/tex]
3. Use the arc length formula: The arc length [tex]\( L \)[/tex] traced by the minute hand is given by:
[tex]\[
L = r \times \theta
\][/tex]
where [tex]\( r = 1.5 \)[/tex] cm (length of the minute hand) and [tex]\( \theta = \frac{\pi}{2} \)[/tex].
4. Calculate the arc length:
[tex]\[
L = 1.5 \times \frac{\pi}{2} = \frac{1.5\pi}{2}
\][/tex]
5. Find the arc length moved: This gives us an arc length of approximately 2.36 cm.
So, for the two questions, the radius of the circle is about 35.71 cm, and the arc length moved by the minute hand is approximately 2.36 cm.
### Question 1
Find the radius of the circle where a central angle of 60° intercepts an arc of length 37.4 cm.
1. Understand the relationship: The length of an arc is related to the angle it subtends at the center of the circle and the radius of the circle. The formula to find the arc length [tex]\( L \)[/tex] is given by:
[tex]\[
L = r \times \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the central angle in radians.
2. Convert the angle to radians: The angle is given in degrees, so we need to convert it to radians. The conversion factor is [tex]\(\pi\)[/tex] radians = 180 degrees. For a 60° angle, the conversion is:
[tex]\[
\theta = 60^\circ \times \left(\frac{\pi}{180}\right) = \frac{\pi}{3} \text{ radians}
\][/tex]
3. Plug into the formula and solve for the radius:
- Given: [tex]\( L = 37.4 \)[/tex] cm
- Substitute the known values into the arc length formula:
[tex]\[
37.4 = r \times \frac{\pi}{3}
\][/tex]
4. Rearrange to find the radius:
[tex]\[
r = \frac{37.4}{\frac{\pi}{3}} = \frac{37.4 \times 3}{\pi}
\][/tex]
5. Calculate the radius: This gives us a radius of approximately 35.71 cm.
### Question 2
Calculate the arc length the minute hand of a watch moves in 15 minutes, where the minute hand is 1.5 cm long.
1. Understand the movement of the minute hand: The minute hand of a watch completes a full revolution (360°) in 60 minutes. Therefore, in 15 minutes, the angle it covers is:
[tex]\[
\frac{15}{60} \times 360^\circ = 90^\circ
\][/tex]
2. Convert the angle to radians:
[tex]\[
90^\circ \times \left(\frac{\pi}{180}\right) = \frac{\pi}{2} \text{ radians}
\][/tex]
3. Use the arc length formula: The arc length [tex]\( L \)[/tex] traced by the minute hand is given by:
[tex]\[
L = r \times \theta
\][/tex]
where [tex]\( r = 1.5 \)[/tex] cm (length of the minute hand) and [tex]\( \theta = \frac{\pi}{2} \)[/tex].
4. Calculate the arc length:
[tex]\[
L = 1.5 \times \frac{\pi}{2} = \frac{1.5\pi}{2}
\][/tex]
5. Find the arc length moved: This gives us an arc length of approximately 2.36 cm.
So, for the two questions, the radius of the circle is about 35.71 cm, and the arc length moved by the minute hand is approximately 2.36 cm.
Thanks for taking the time to read Exercise 4 3 1 Find the radius of the circle in which a central angle of 60 intercepts an arc of length 37 4 cm. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
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Rewritten by : Barada
