Math Problem Statement

If EG has a length of 6π inches, find the radius of the circle.

Solution

To solve this problem, we need to determine the length EGEG, which is given as 6π6 \pi inches. The diagram shows a circle with center OO, radius OE=OFOE = OF, and EOF=30\angle EOF = 30^\circ. Since EGEG is an arc spanning this angle, we'll interpret 6π6 \pi inches as the arc length of EGEG over 3030^\circ.

Step-by-Step Solution:

  1. Identify the Arc Length Formula: The arc length ss of a circle is given by: s=rθs = r \theta where rr is the radius and θ\theta is the angle in radians.

  2. Convert Degrees to Radians: Since EOF=30\angle EOF = 30^\circ, we convert this to radians: θ=30×π180=π6\theta = 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6}

  3. Set Up the Equation: We know that s=6πs = 6 \pi and θ=π6\theta = \frac{\pi}{6}. Plugging these into the arc length formula: 6π=rπ66 \pi = r \cdot \frac{\pi}{6}

  4. Solve for the Radius rr: Divide both sides by π6\frac{\pi}{6} to isolate rr: r=6ππ6=6×6=36 inchesr = \frac{6 \pi}{\frac{\pi}{6}} = 6 \times 6 = 36 \text{ inches}

  5. Confirm the Answer: The radius OEOE (or OFOF) of the circle is 3636 inches.

Final Answer:

The radius OE=OF=36OE = OF = 36 inches.

Would you like more details on any of these steps or have any further questions?

Here are some related questions to explore:

  1. What is the formula for converting degrees to radians?
  2. How do you derive the arc length formula s=rθs = r \theta?
  3. If the angle EOF\angle EOF were 6060^\circ, what would the arc length EGEG be?
  4. How would the solution change if EGEG spanned 9090^\circ instead of 3030^\circ?
  5. Can you explain why dividing by π6\frac{\pi}{6} in step 4 is the correct approach?

Tip: Remember that angles in radians make many formulas in trigonometry and geometry simpler to apply, especially for arc lengths and sector areas.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Properties
Radians and Arc Length

Formulas

Arc Length Formula s = rθ
Degree to Radian Conversion

Theorems

Central Angle and Arc Length Relationship

Suitable Grade Level

Grades 9-10