Question

100 POINTS!!!! Please solve this

Answer 1

58.06 ft

Explanation:

`\boxed{\left\begin{array}{ccc}\text{\underline{Length of a Sector(Arc Length):}}\\\\L=(\theta)/(180 \textdegree) \pi r \end{array}\right \left\begin{array}{ccc}\text{\underline{Circumference of a Circle:}}\\\\C=2\pi r\end{array}\right }`

Given:

`L_(DE)=46.75 \ ft\\\\\theta=290 \textdegree`

Find:

`C=?? \ ft`

(1) - Use the information we know about sector DE to find the radius of the circle, "r"

`L_(DE)=(\theta)/(180 \textdegree) \pi r \\\\\Longrightarrow 46.75=(290 \textdegree)/(180 \textdegree)\pi r\\ \\ \Longrightarrow 46.75=(29)/(18)\pi r\\\\\Longrightarrow r=46.75(18)/(29\pi)\\ \\\therefore \boxed{r\approx 9.24 \ ft}`

(2) - Use the value we just found for r and use it to find the circumference of the circle

`C=2 \pi r\\\\\Longrightarrow C=2 \pi (9.24)\\\\\therefore \boxed{\boxed{C\approx 58.06 \ ft}}`

Thus, the circumference of the circle is found.

Answer 2

58.03 ft

Explanation:

Given an arc of 290° has a length of 46.75 ft, you want to know the circumference of the circle.

Arc length

Arc length is proportional to the central angle. For a circle of the same radius, an arc of 360° (the whole circle) will have a length that is 360/290 times the arc with an angle of 290°.

(46.75 ft) × 360°/290° ≈ 58.03 ft

The circumference of circle F is about 58.03 feet.

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Additional comment

The arc length is given by the formula

s = rθ . . . . where r is the radius and θ is the central angle in radians

We could go to the trouble to find the angle in radians and the radius of the circle, but that is not necessary. This tells us that the arc length is proportional to the angle, which is all we need to know to solve this problem.

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