Name: $\qquad$ Grace R $\newline$ $\mathrm{col}$ $\newline$ $10$ . Bob is running for class president and is making signs to put up on a large column outside his school's gymnasium. He wants to make sure the text can be seen in its entirety when students are exiting the gym's front door. The diagram shows an overhead views of the circular column near the front door. The lines of sight to the column make a $32^{\circ}$ angle. What is the largest portion of the column his sign covers if he wants to make sure students can see the whole text from the front door?

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Q. Name:

\qquad

Grace R

\newline

\mathrm{col}

\newline

10

. Bob is running for class president and is making signs to put up on a large column outside his school's gymnasium. He wants to make sure the text can be seen in its entirety when students are exiting the gym's front door. The diagram shows an overhead views of the circular column near the front door. The lines of sight to the column make a

32^{\circ}

angle. What is the largest portion of the column his sign covers if he wants to make sure students can see the whole text from the front door?

Calculate Circumference: Calculate the circumference of the column since the sign will wrap around it. $\newline$ Circumference formula: $C = 2 \cdot \pi \cdot r$ $\newline$ We don't have the radius, but we can use the angle to find the portion of the circumference.
Angle of Sight: The angle of sight is $32$ degrees, which is the fraction of the full $360$ degrees around the column. $\newline$ So, the fraction of the circumference that the sign can cover is $\frac{32}{360}.$
Apply Fraction to Formula: Now, we need to apply this fraction to the circumference formula to find the length of the sign. $\newline$ Length of sign = $(\frac{32}{360}) \times C$ $\newline$ But we don't have the actual circumference, so we can't calculate the exact length.
Express Length in Terms of Radius: Since we can't calculate the exact length without the radius, we express the length of the sign in terms of the radius. $\newline$ Length of sign = $(\frac{32}{360}) \times (2 \times \pi \times r)$ $\newline$ Simplify the expression. $\newline$ Length of sign = $(\frac{32}{360}) \times 2\pi r$ $\newline$ Length of sign = $(\frac{16}{180}) \times 2\pi r$ $\newline$ Length of sign = $(\frac{2}{45}) \times 2\pi r$ $\newline$ Length of sign = $(\frac{4\pi r}{45})$
Correct Simplification Error: We made a mistake in the simplification step; it should be $(\frac{32}{360}) \times 2\pi r = (\frac{1}{11.25}) \times 2\pi r$ . Correct the error. Length of sign = $(\frac{1}{11.25}) \times 2\pi r$ Length of sign = $(\frac{2\pi r}{11.25})$