Answer :
To solve this question, we need to tackle it in two parts. Let's break down each part step-by-step.
Part 1: Finding the radius of the circle
Given:
- Arc length = 176 cm
- Angle subtended at the center = 180°
- [tex]\pi[/tex] ≈ [tex]\frac{22}{7}[/tex]
The formula to find the arc length ([tex]L[/tex]) is:
[tex]L = \frac{\theta}{360} \times 2 \pi r[/tex]
where [tex]\theta[/tex] is the angle in degrees, [tex]r[/tex] is the radius of the circle, and [tex]\pi[/tex] is approximately [tex]\frac{22}{7}[/tex].
Insert the given values:
[tex]176 = \frac{180}{360} \times 2 \times \frac{22}{7} \times r[/tex]
Simplify the equation:
[tex]176 = \frac{1}{2} \times \frac{44}{7} \times r[/tex]
[tex]176 = \frac{22}{7} \times r[/tex]
To find [tex]r[/tex], multiply both sides by [tex]\frac{7}{22}[/tex]:
[tex]r = 176 \times \frac{7}{22}[/tex]
[tex]r = 56 \text{ cm}[/tex]
So, the radius of the circle is 56 cm.
Part 2: Distance covered by the tip of the minute hand in 45 minutes
Given:
- Length of the minute hand = 5.6 cm
- [tex]\pi[/tex] ≈ 3.142
The circumference of the full circle that the minute hand would cover in 60 minutes is:
[tex]C = 2 \pi r[/tex]
Substitute [tex]r[/tex] = 5.6 cm:
[tex]C = 2 \times 3.142 \times 5.6[/tex]
[tex]C = 35.1896 \text{ cm}[/tex]
Since the question asks for the distance covered in 45 minutes:
Since 45 minutes is [tex]\frac{3}{4}[/tex] of an hour, the distance covered is [tex]\frac{3}{4}[/tex] of the circumference:
[tex]D = \frac{3}{4} \times 35.1896[/tex]
[tex]D \approx 26.392 \text{ cm}[/tex]
Thus, the tip of the minute hand covers approximately 26.392 cm in 45 minutes.
