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Part A: What angle in radians is subtended by an arc of 1.49 mm in length on the circumference of a circle with a radius of 2.46 mm? Express your answer in radians.

Part B: What is this angle in degrees? Express your answer in degrees.

Part C: An arc of length 14.5 cm on the circumference of a circle subtends an angle of 121 degrees. What is the radius of the circle? Express your answer in centimeters.

Part D: The angle between two radii of a circle with a radius of 1.53 mm is 0.720 radians. What length of arc is intercepted on the circumference of the circle by the two radii? Express your answer in meters.

Answer :

Let's go through each part of your question step-by-step:

Part A:

To find the angle in radians subtended by an arc, we use the formula:

[tex]\theta = \frac{s}{r}[/tex]

where:

  • [tex]\theta[/tex] is the angle in radians,
  • [tex]s[/tex] is the arc length,
  • [tex]r[/tex] is the radius of the circle.

Given:

  • Arc length [tex]s = 1.49[/tex] mm
  • Radius [tex]r = 2.46[/tex] mm

Substitute these into the formula:

[tex]\theta = \frac{1.49}{2.46} \approx 0.6057\, \text{radians}[/tex]

Part B:

To convert an angle given in radians to degrees, use:

[tex]\text{Degrees} = \text{Radians} \times \frac{180}{\pi}[/tex]

Using the angle from Part A:

[tex]\text{Degrees} = 0.6057 \times \frac{180}{\pi} \approx 34.714\, \text{degrees}[/tex]

Part C:

Here, we need to find the radius of the circle when given an arc length and the subtended angle in degrees.
The formula connecting these is:

[tex]s = r \theta[/tex]

First, convert the angle from degrees to radians:

[tex]\theta = 121 \times \frac{\pi}{180} \approx 2.1118 \text{ radians}[/tex]

Using the formula with arc length [tex]s = 14.5\, \text{cm}[/tex] and [tex]\theta = 2.1118\, \text{radians}[/tex]:

[tex]14.5 = r \cdot 2.1118[/tex]

Solve for [tex]r[/tex]:

[tex]r = \frac{14.5}{2.1118} \approx 6.86\, \text{cm}[/tex]

Part D:

To find the arc length when the angle and the radius are known, use the same formula:

[tex]s = r \theta[/tex]

Given:

  • Radius [tex]r = 1.53[/tex] mm = 0.00153 m (converted to meters)
  • Angle [tex]\theta = 0.720[/tex] radians

[tex]s = 0.00153 \times 0.720 = 0.0011016 \text{ meters}[/tex]

Therefore, the length of the arc is approximately [tex]0.0011016[/tex] meters.

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Rewritten by : Jeany

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