John calculates an arc length involving a circle with a radius of 9 inches. The associated central angle has a measure of [tex]$3 \pi$[/tex] radians.
John's Work
Step 1: Central angle (radians) = [tex]$\frac{\text { arc length }}{\text { radius }}$[/tex]
Step 2: [tex]$3 \pi=\frac{\text { arc length }}{9}$[/tex]
Step 3: arc length = [tex]$\frac{\pi}{3}$[/tex]
John made his first error in step ____. He ____, but he should have ____.
Answer (2)
The formula relating central angle, arc length, and radius is correctly stated.
The values are correctly substituted into the formula.
The arc length is calculated by multiplying the central angle by the radius: 3 π × 9 = 27 π .
John's error was dividing by 9 instead of multiplying, leading to an incorrect arc length of 3 π .
Explanation
Problem Analysis Let's analyze John's work step by step to identify the error in calculating the arc length. We are given that the radius of the circle is 9 inches and the central angle is 3 π radians.
Check Step 1 Step 1 states the relationship between the central angle, arc length, and radius: Central angle (radians) = radius arc length This formula is correct.
Check Step 2 Step 2 substitutes the given values into the formula: 3 π = 9 arc length This substitution is also correct.
Identify the Error Step 3 calculates the arc length. To find the arc length, we need to multiply both sides of the equation in Step 2 by the radius, which is 9 inches: arc length = 3 π × 9 = 27 π John incorrectly calculated the arc length as 3 π . He divided by 9 instead of multiplying by 9. Therefore, John made his first error in Step 3.
Conclusion John made his first error in step 3. He divided by 9, but he should have multiplied by 9.
Examples
Understanding arc length is crucial in many real-world applications. For instance, when designing a curved race track, engineers need to calculate the length of the curves to determine the total distance of the race. Similarly, in manufacturing, calculating the arc length is essential when creating curved components for machines or structures. Also, calculating arc length helps navigators determine distances on maps and globes, which are based on circular measurements of the Earth.
John made his first error in step 3, where he incorrectly divided by 9 instead of multiplying by 9. He should have calculated the arc length using the formula as arc length = 3 π × 9 = 27 π . Thus, the correct arc length is 27 π inches, not 3 π .
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