1) 18 If we create a graph using distance from the crime scene as y and hours since 11 PM as x, we have a point at (5, 10) since 5 hours after 11, he was 10 miles away. We can also use the point (0, 0) since 0 hours after 11, he was 0 miles away. The slope of the line that goes through these two points is (10-0)/(5-0) = 2. The y-intercept is 0 since we know that y=0 at x=0. Therefore his movement is described by the equation y = 2x. At 8 am, the x-value is 9. By plugging in 9 we get y = 2(9) = 18 miles. 2) (9, 2) and (9, 18) It is possible that he ends up running in the same direction, away from the crime. In this case he would end up 18 miles, at the point (9,18). However, he may have started running back to the crime scene after 4 AM. In this case, his position must be graphed with a slope of -2. Create a line with a slope of -2 that goes through the point (5, 10) and then find the distance (the y value) at 8 AM (x = 9). This could also be solved algebraically by solving for the y-intercept. 10 = -2(5) + b 20 = b y = -2x + 20 At 8am: y = -2(9) + 20 = 2 Therefore the two solutions are (9, 18) and (9, 2) 3) (25, 47) It has been 2.5 hours since 8am. As Jack moves at a rate of 2 miles per hour, using the equation Distance = Time x Rate we can see that he has gone a distance of 2 x 2.5 = 5 miles. Using the distance formula for each of the given coordinates we select the one that has a distance of 5 miles from the original point of (22, 43). (25, 47): Sqrt[ (25-22)^2 + (47-43)^2 ] = 5 (23, 31): Sqrt[ (23-22)^2 + (31-43)^2 ] = 8.06 (19, 41): Sqrt[ (19-22)^2 + (41-43)^2 ] = 3.61 (27, 46): Sqrt[ (27-22)^2 + (43-46)^2 ] = 5.83 4) All points on the circle will be the fastest because they’re all equidistant from the center Since all of the Starbucks are found in a map on a circle with radius 5 and its center on the point (25, 47), they will all be a distance of 5 miles away and take equally long to get to.