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Answer :
We solve the problem in two parts.
─────────────────────────────
Part 1.
We are given that a central angle of [tex]$60^\circ$[/tex] intercepts an arc of length [tex]$37.4\text{ cm}$[/tex], and we are to use [tex]$\pi=\frac{22}{7}$[/tex]. The formula for the length of an arc is
[tex]$$
s = r \theta,
$$[/tex]
where [tex]$s$[/tex] is the arc length, [tex]$r$[/tex] is the radius of the circle, and [tex]$\theta$[/tex] is the central angle in radians.
1. First, convert the central angle from degrees to radians. We use the relation
[tex]$$
\theta = 60^\circ \times \left(\frac{\pi}{180}\right).
$$[/tex]
Substituting [tex]$\pi=\frac{22}{7}$[/tex], we have
[tex]$$
\theta = 60 \times \left(\frac{22/7}{180}\right) = 60 \times \frac{22}{7 \times 180} = \frac{60 \times 22}{1260} = \frac{1320}{1260} = \frac{22}{21} \, \text{radians}.
$$[/tex]
This is approximately
[tex]$$
\theta \approx 1.04762 \, \text{radians}.
$$[/tex]
2. With the arc length [tex]$s = 37.4\text{ cm}$[/tex] and the computed [tex]$\theta$[/tex], we solve for the radius [tex]$r$[/tex]:
[tex]$$
r = \frac{s}{\theta} = \frac{37.4}{\frac{22}{21}} = 37.4 \times \frac{21}{22}.
$$[/tex]
Evaluating the above expression gives
[tex]$$
r \approx 35.70\text{ cm}.
$$[/tex]
─────────────────────────────
Part 2.
The minute hand of a watch has a length of [tex]$1.5$[/tex] (this is the radius of the circle traced by its tip). The full circumference of the circle is
[tex]$$
C = 2\pi r = 2\pi (1.5).
$$[/tex]
Here, [tex]$\pi$[/tex] is taken as the usual mathematical constant (approximately [tex]$3.14159$[/tex]). Thus, the circumference is
[tex]$$
C \approx 2 \times 3.14159 \times 1.5 \approx 9.42478.
$$[/tex]
Since in 15 minutes the minute hand covers [tex]$\frac{15}{60} = \frac{1}{4}$[/tex] of the circle, the distance traveled by the tip is
[tex]$$
\text{Distance} = \frac{1}{4} \times C \approx \frac{1}{4} \times 9.42478 \approx 2.35619.
$$[/tex]
─────────────────────────────
Final Answers:
1. The radius of the circle is approximately [tex]$35.70\text{ cm}$[/tex].
2. The tip of the minute hand moves approximately [tex]$2.35619\text{ cm}$[/tex] in 15 minutes.
─────────────────────────────
Part 1.
We are given that a central angle of [tex]$60^\circ$[/tex] intercepts an arc of length [tex]$37.4\text{ cm}$[/tex], and we are to use [tex]$\pi=\frac{22}{7}$[/tex]. The formula for the length of an arc is
[tex]$$
s = r \theta,
$$[/tex]
where [tex]$s$[/tex] is the arc length, [tex]$r$[/tex] is the radius of the circle, and [tex]$\theta$[/tex] is the central angle in radians.
1. First, convert the central angle from degrees to radians. We use the relation
[tex]$$
\theta = 60^\circ \times \left(\frac{\pi}{180}\right).
$$[/tex]
Substituting [tex]$\pi=\frac{22}{7}$[/tex], we have
[tex]$$
\theta = 60 \times \left(\frac{22/7}{180}\right) = 60 \times \frac{22}{7 \times 180} = \frac{60 \times 22}{1260} = \frac{1320}{1260} = \frac{22}{21} \, \text{radians}.
$$[/tex]
This is approximately
[tex]$$
\theta \approx 1.04762 \, \text{radians}.
$$[/tex]
2. With the arc length [tex]$s = 37.4\text{ cm}$[/tex] and the computed [tex]$\theta$[/tex], we solve for the radius [tex]$r$[/tex]:
[tex]$$
r = \frac{s}{\theta} = \frac{37.4}{\frac{22}{21}} = 37.4 \times \frac{21}{22}.
$$[/tex]
Evaluating the above expression gives
[tex]$$
r \approx 35.70\text{ cm}.
$$[/tex]
─────────────────────────────
Part 2.
The minute hand of a watch has a length of [tex]$1.5$[/tex] (this is the radius of the circle traced by its tip). The full circumference of the circle is
[tex]$$
C = 2\pi r = 2\pi (1.5).
$$[/tex]
Here, [tex]$\pi$[/tex] is taken as the usual mathematical constant (approximately [tex]$3.14159$[/tex]). Thus, the circumference is
[tex]$$
C \approx 2 \times 3.14159 \times 1.5 \approx 9.42478.
$$[/tex]
Since in 15 minutes the minute hand covers [tex]$\frac{15}{60} = \frac{1}{4}$[/tex] of the circle, the distance traveled by the tip is
[tex]$$
\text{Distance} = \frac{1}{4} \times C \approx \frac{1}{4} \times 9.42478 \approx 2.35619.
$$[/tex]
─────────────────────────────
Final Answers:
1. The radius of the circle is approximately [tex]$35.70\text{ cm}$[/tex].
2. The tip of the minute hand moves approximately [tex]$2.35619\text{ cm}$[/tex] in 15 minutes.
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