Question

a spinner with 18 equal sectors starts as shown below. the arrow is in the middle of the top section. it is spun clockwise, but doesnt even rotate a whole turn, stopping in the middle of section 15. how many degrees does the arrow rotate?

Explanation:

Step1: Find degrees per section

A full circle is \(360^\circ\), and there are 18 equal sections. So degrees per section: \(\frac{360^\circ}{18} = 20^\circ\) per section.

Step2: Determine number of sections moved

Starts at section 0 (top), ends at section 15. Moving clockwise, number of sections moved: from 0 to 15, that's 15 - 0 = 15? Wait, no. Wait, let's count the sections. Wait, the spinner has sections 0,1,2,...,17. Wait, starting at 0 (top), moving clockwise, section 1 is next, then 2, ..., 15. Wait, how many sections between 0 and 15? Wait, no, maybe I got the direction wrong. Wait, clockwise: from 0, clockwise, the next section is 1? Wait, no, looking at the diagram: the top is 0, then clockwise (right) is 1, 2, ..., then down, then left? Wait, no, the diagram: 0 is top, then 1 is to the right of 0, 2 to the right of 1, ..., then 15 is on the left side. Wait, maybe the numbering is clockwise: 0 (top), 1 (right-top), 2 (right), ..., 8 (bottom-right), 9 (bottom), 10 (bottom-left), 11 (left-bottom), 12 (left), 13 (left-top), 14 (top-left), 15 (top-left? Wait, no, the diagram shows 15 is on the left, 16 above 15, 17 above 16, then 0. Wait, maybe the sections are numbered clockwise: 0 (top), 1 (clockwise next), 2, ..., 17, then back to 0. Wait, but when moving clockwise from 0, how many sections to get to 15? Wait, 0 to 1: 1 section, 0 to 2: 2, ..., 0 to 15: 15 sections? Wait, no, maybe I made a mistake. Wait, let's check the total sections: 18. So each section is \(20^\circ\). Now, starting at 0 (middle of top section), ending at middle of section 15. Let's count the number of sections moved. Let's list the sections in clockwise order: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, then back to 0. Wait, no, 18 sections: 0-17. So from 0 to 15, moving clockwise: how many steps? 0 to 1: 1, 0 to 2: 2, ..., 0 to 15: 15? Wait, no, 0 to 15: 15 sections? Wait, but 18 sections, so 0 to 15 is 15 sections? Wait, no, 0 to 1 is 1, 0 to 17 is 17 sections (clockwise). Wait, maybe the numbering is counter-clockwise? No, the problem says "spun clockwise". Wait, maybe I got the direction of numbering wrong. Wait, the arrow is in the middle of the top section (0). Spun clockwise, stops in middle of section 15. Let's count the number of sections between 0 and 15 when moving clockwise. Let's see the diagram: 0 is top, 1 is right of 0, 2 right of 1, ..., 8 is bottom, 9 is bottom-left, 10 is left-bottom, 11 is left, 12 is left-top, 13 is top-left, 14 is above 13, 15 is above 14, 16 above 15, 17 above 16, then 0. Wait, no, that can't be. Wait, maybe the sections are numbered clockwise: 0 (top), 1 (clockwise, so to the right), 2, 3, ..., 8 (bottom), 9 (clockwise from 8, so bottom-right? No, bottom is 9? Wait, the diagram shows 9 at the bottom. So 0 (top), 1 (right-top), 2 (right), 3 (right-bottom), 4 (bottom-right), 5 (bottom), 6 (bottom-left), 7 (left-bottom), 8 (left), 9 (left-top), 10 (top-left), 11 (above 10), 12 (above 11), 13 (above 12), 14 (above 13), 15 (above 14), 16 (above 15), 17 (above 16), then back to 0. Wait, no, that's not 18 sections. Wait, the problem says 18 equal sectors. So 360/18=20 degrees per sector. Now, starting at 0 (top), moving clockwise to 15. Let's count the number of sectors moved. From 0 to 15: how many sectors? Let's see the numbers: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17. So 0 to 15: 15 sectors? Wait, no, 0 to 1 is 1, 0 to 2 is 2, ..., 0 to 15 is 15? Wait, but 15 sectors would be 15*20=300 degrees. But wait, maybe I made a mistake. Wait, maybe the direction is counter-clockwise? No, the problem says clockwise. Wait, another approach: the total number of sectors between 0 and 15 when moving clockwise. Wait, 0 to 15: 15 sectors? Wait, 0 to 15: 15 steps? Wait, 0 to 1: 1 step, 0 to 15: 15 steps. So 15*20=300 degrees. Wait, but let's check: 18 sectors, 20 degrees each. If we move from 0 to 15 clockwise, that's 15 sectors. So 15*20=300 degrees. Alternatively, maybe the numbering is such that 0 is top, and 15 is three sectors before 0 (since 18-3=15). Wait, no, 18 sectors: 0,1,...,17. So 0 to 15 clockwise: 15 sectors. So 15*20=300 degrees.

Answer:

\(300^\circ\)