Math Problem Statement
Solution
To solve this problem, we need to determine the length , which is given as inches. The diagram shows a circle with center , radius , and . Since is an arc spanning this angle, we'll interpret inches as the arc length of over .
Step-by-Step Solution:
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Identify the Arc Length Formula: The arc length of a circle is given by: where is the radius and is the angle in radians.
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Convert Degrees to Radians: Since , we convert this to radians:
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Set Up the Equation: We know that and . Plugging these into the arc length formula:
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Solve for the Radius : Divide both sides by to isolate :
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Confirm the Answer: The radius (or ) of the circle is inches.
Final Answer:
The radius inches.
Would you like more details on any of these steps or have any further questions?
Here are some related questions to explore:
- What is the formula for converting degrees to radians?
- How do you derive the arc length formula ?
- If the angle were , what would the arc length be?
- How would the solution change if spanned instead of ?
- Can you explain why dividing by 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
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