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$If the minute hand of a clock is 3.5 inches long and its tip rotates through a distance of 8 inches, then which of the following is closest to the angle that it rotates? (1) 131^(@) (3) 267^(@) {:[theta=(5)/(r)],[(8)/(3.5)((180 )/(pi))]:} (2) 174^(@) (4) 314^(@)$

$7$ . If the minute hand of a clock is $3$ . $5$ inches long and its tip rotates through a distance of $8$ inches, then which of the following is closest to the angle that it rotates? $\newline$ ( $1$ ) $131^{\circ}$ $\newline$ ( $3$ ) $267^{\circ}$ $\newline$ $\begin{array}{l} \theta=\frac{5}{r} \\ \frac{8}{3.5}\left(\frac{180}{\pi}\right) \end{array}$ $\newline$ ( $2$ ) $174^{\circ}$ $\newline$ ( $4$ ) $314^{\circ}$

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7

. If the minute hand of a clock is

3

5

inches long and its tip rotates through a distance of

8

inches, then which of the following is closest to the angle that it rotates?

\newline

(

1

)

131^{\circ}

\newline

(

3

)

267^{\circ}

\newline

\begin{array}{l} \theta=\frac{5}{r} \\ \frac{8}{3.5}\left(\frac{180}{\pi}\right) \end{array}

\newline

(

2

)

174^{\circ}

\newline

(

4

)

314^{\circ}

Use Circle Arc Length Formula: First, we need to use the formula for the arc length of a circle, which is $\theta = \frac{\text{arc length}}{\text{radius}}$ . Here, the arc length is $8$ inches and the radius is the length of the minute hand, which is $3.5$ inches.
Calculate Theta: So, we plug in the numbers: $\theta = \frac{8}{3.5}$ .
Convert to Degrees: Calculating that gives us $\theta = 2.2857\ldots$ (rounded off it's about $2.29$ ).
Correct Radians Calculation: Now, we need to convert this into degrees because the answer choices are in degrees. To convert from radians to degrees, we multiply by $(180/\pi)$ .
Correct Radians Calculation: Now, we need to convert this into degrees because the answer choices are in degrees. To convert from radians to degrees, we multiply by $(180/\pi)$ .So, we do $2.29 \times (180/\pi)$ . But wait, I just realized we didn't actually calculate the radians correctly. We forgot to include $\pi$ in the original formula. The correct formula should be $\theta = (\text{arc length} / \text{radius}) \times (180/\pi)$ . Let's fix that.