Math Problem Statement

ind the radius r of the circle if an arc of length 6 m on the circle subtends a central angle of 3𝜋/7 rad. (Round your answer to two decimal places.)

Solution

To find the radius rr of the circle, you can use the formula relating the arc length ss, the radius rr, and the central angle θ\theta (in radians):

s=rθs = r \theta

Given:

  • Arc length, s=6s = 6 m
  • Central angle, θ=3π7\theta = \frac{3\pi}{7} rad

We need to find the radius rr. Rearrange the formula to solve for rr:

r=sθr = \frac{s}{\theta}

Substitute the values:

r=63π7=6×73π=423πr = \frac{6}{\frac{3\pi}{7}} = \frac{6 \times 7}{3\pi} = \frac{42}{3\pi}

Now, calculate rr and round to two decimal places:

r=423π429.42484.46 mr = \frac{42}{3\pi} \approx \frac{42}{9.4248} \approx 4.46 \text{ m}

So, the radius rr is approximately 4.46 meters.

Would you like further details or have any questions?

Here are 5 related questions for you to explore:

  1. How would the radius change if the central angle were doubled?
  2. What is the formula for arc length if the angle is given in degrees?
  3. How do you convert a central angle from radians to degrees?
  4. What is the relationship between the circumference of a circle and the central angle for a full circle?
  5. How would the radius change if the arc length were halved?

Tip: Always ensure the central angle is in radians when using the arc length formula.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Geometry

Formulas

Arc length formula: s = r * theta

Theorems

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Suitable Grade Level

High School