Lin and Elena are trying to find the area of the shaded sector in the image.

Lin says, “We’ve found sector areas when the central angles are given in degrees, but here the central angle is in radians. Should I start by finding the area of the full circle?”

Elena says, “I saw someone using the formula , where is the measure of the angle in radians, and is the radius. But I don’t know where that came from.”

1. Compare and contrast finding sector areas for central angles measured in degrees and those measured in radians.
2. Explain why the formula that Elena saw works.
3. Find the area of the sector.

A city is designing a new park. One feature of the park is a garden designed for walking meditation. The space they have chosen is 14 yards long on one edge. There will be alternating circular arcs of plants and paved paths.

Image of the city plan for a new park shaped like circular arcs. From point T, two straight lines extend in two directions and opening to the right. Horizontal line T D has points A, B, C and D. The length of segment T A is 2 yards, A B is 4 yards, B C is 5 yards and C D is 3 yards. At points A. B, C and D, arcs extend up to the left to the other straight line. The arc length at A is 3 yards, at B is 9 yards, at C is 16 point 5 yards and the arc length at D is 21 yards.

1. Create a table that shows the arc length  of the curved paths as a function of the radius  of the straight connecting paths.
2. Plot the points from your table on the coordinate grid, and connect them.

   

3. The points should form a line. Write an equation for this line, using the variables and .
4. What does the slope of the line mean in the context of the curved and connecting paths?
5. The city is considering building another curved path starting at point  that will be an additional 6 yards past point . How far would the walk be from one straight connector path to the other?