Let's go through each step of John's work and identify where he made the error.
1. Step 1:
[tex]\[
\text{Central angle (radians)} = \frac{\text{arc length}}{\text{radius}}
\][/tex]
This step correctly shows how the central angle relates to the arc length and radius.
2. Step 2:
[tex]\[
3\pi = \frac{\text{arc length}}{9}
\][/tex]
This equation correctly sets the central angle given as [tex]\(3\pi\)[/tex] radians equal to the arc length divided by the radius of 9 inches.
3. Step 3:
[tex]\[
\text{arc length} = \frac{\pi}{3}
\][/tex]
This is where John made his first error. He should have multiplied the central angle by the radius to find the arc length.
4. The correct calculation for arc length should be:
[tex]\[
\text{arc length} = 3\pi \times 9 = 27\pi \text{ inches}
\][/tex]
Since we are told the correct arc length is [tex]\(84.82300164692441\)[/tex] inches (which is equivalent to [tex]\(27\pi\)[/tex]), we need to fill in the missing parts:
John made his first error in step: 3.
He: calculated the arc length incorrectly,
but he should have: multiplied the central angle by the radius to find the arc length.
In summary, the filled sentence should read:
"John made his first error in step 3. He calculated the arc length incorrectly, but he should have multiplied the central angle by the radius to find the arc length."
