To determine the length of the track around turn B, we need to use the formula for the length of an arc of a circle. The formula is:
[tex]\[ \text{Arc Length} = \left(\frac{\text{central angle}}{360}\right) \times 2 \pi \times \text{radius} \][/tex]
From question 2, we know the central angle ([tex]\(\angle B\)[/tex]) is [tex]\(102^\circ\)[/tex] and the radius of the circle is given as [tex]\(9\)[/tex] miles. Here, [tex]\(\pi\)[/tex] (Pi) is a constant approximately equal to 3.14159.
Step-by-step solution:
1. Substitute the values into the arc length formula:
[tex]\[
\text{Arc Length} = \left(\frac{102}{360}\right) \times 2 \pi \times 9
\][/tex]
2. Simplify the fraction:
[tex]\[
\frac{102}{360} = \frac{51}{180} = \frac{17}{60}
\][/tex]
3. Substitute back into the formula:
[tex]\[
\text{Arc Length} = \left(\frac{17}{60}\right) \times 2 \pi \times 9
\][/tex]
4. Multiply the constant terms:
[tex]\[
\text{Arc Length} = \left(\frac{17}{60}\right) \times 18 \pi
\][/tex]
5. Further simplify this:
[tex]\[
\left(\frac{17}{60}\right) \times 18 = \frac{17 \times 18}{60} = \frac{306}{60} = 5.1
\][/tex]
6. Now use this with [tex]\(\pi\)[/tex]:
[tex]\[
\text{Arc Length} = 5.1 \pi
\][/tex]
7. Multiply by [tex]\(\pi = 3.14159\)[/tex] to find the numerical arc length:
[tex]\[
\text{Arc Length} = 5.1 \times 3.14159 \approx 16.0221
\][/tex]
So, the length of the track around turn B is approximately [tex]\(16.0221\)[/tex] miles. Rounding to three decimal places, we get:
[tex]\[ \text{Arc Length} \approx 16.022 \][/tex]
Therefore, the length of the track around turn B is [tex]\(16.022\)[/tex] miles.
