Question

Use the arc length formula and the given information to find θ. s = 4 m, r = 13 m; θ = ?

1) 52 rad
2) four divided by thirteen rad
3) 104 rad
4) thirteen divided by four rad

Answer 1

The arc length formula and the given information to find θ. s = 4 m, r = 13 m; θ =

`\[ \theta = 4 \, \text{rad} \]`

Step-by-step explanation:

The arc length formula is given by
`\(s = r \theta\), where \(s\)` is the arc length,
`\(r\) is the radius, and \(\theta\)` is the central angle in radians. In this case,
`\(s = 4 \, \text{m}\) and \(r = 13 \, \text{m}\). To find \(\theta\),` rearrange the formula to solve for
`\(\theta\):`

`\[ \theta = (s)/(r) \]`

Substitute the given values:

`\[ \theta = \frac{4 \, \text{m}}{13 \, \text{m}} \]`

Simplify the fraction:

`\[ \theta = (4)/(13) \, \text{rad} \]`

Thus, the correct answer is
`\( \theta = (4)/(13) \, \text{rad} \),` which corresponds to the option "four divided by thirteen rad" in the choices provided.

This result makes sense intuitively – the angle is a fraction of the full circle, indicating that the arc length of 4 meters is a small portion of the circumference of a circle with a radius of 13 meters. The answer is consistent with the mathematical relationship between arc length, radius, and central angle in a circle.