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In a circle with a radius of 36.9 m, an arc is intercepted by a central angle of [tex]$\frac{8 \pi}{5}$[/tex] radians.

What is the arc length?

Use 3.14 for [tex]$\pi$[/tex] and round your final answer to the nearest hundredth.

Enter your answer as a decimal in the box:

[tex]\square[/tex] m

Answer :

We are given a circle with a radius of
[tex]$$ r = 36.9 \text{ m} $$[/tex]
and a central angle of
[tex]$$ \theta = \frac{8\pi}{5} \text{ radians}. $$[/tex]

Since the problem instructs us to use
[tex]$$ \pi = 3.14, $$[/tex]
we substitute this value into the central angle:

[tex]$$
\theta = \frac{8 \times 3.14}{5}.
$$[/tex]

Calculating the numerator:

[tex]$$
8 \times 3.14 = 25.12,
$$[/tex]

and then dividing by 5:

[tex]$$
\theta = \frac{25.12}{5} = 5.024 \text{ radians}.
$$[/tex]

The formula for the length of an arc intercepted by a central angle in a circle is:

[tex]$$
\text{Arc Length} = r \times \theta.
$$[/tex]

Substitute the known values into the formula:

[tex]$$
\text{Arc Length} = 36.9 \times 5.024.
$$[/tex]

Multiplying these values:

[tex]$$
\text{Arc Length} = 185.3856 \text{ m}.
$$[/tex]

Finally, rounding this value to the nearest hundredth gives:

[tex]$$
\text{Arc Length} \approx 185.39 \text{ m}.
$$[/tex]

Thus, the length of the arc is

[tex]$$\boxed{185.39 \text{ m}}.$$[/tex]

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