The graph shows the depth, y, in meters, of a shark from the surface of an ocean for a certain amount of time, x, in minutes:
A graph is titled Distance Vs. Time is shown. The x axis is labeled Time in minutes and shows numbers 0, 1, 2, 3, 4, 5. The y axis is labeled Distance from Ocean Surface in meters. A straight line joins the points C at ordered pair 0,66, B at ordered pair 1, 110, A at ordered pair 2, 154, and the ordered pair 3, 198.
Part A: Describe how you can use similar triangles to explain why the slope of the graph between points A and B is the same as the slope of the graph between points A and C. (4 points)
Part B: What are the initial value and slope of the graph, and what do they represent? (6 points)
Part A: In the graph, points A, B, and C form a line. This means that the line passes through these points and has a constant slope between any two points. By looking at the triangles formed by the points A, B, and C, we can see that they are similar triangles. Since similar triangles have the same angles, the ratios of corresponding sides are equal. Therefore, the slope between points A and B is the same as the slope between points A and C.
Part B:
- The initial value of the graph is the y-intercept, which is the value of y when x = 0. In this case, the initial value is 66, which means the shark was 66 meters below the ocean surface at time 0 minutes.
- The slope of the graph represents the rate at which the shark is descending below the ocean surface. In this case, the slope is (110-66)/(1-0) = 44 meters per minute. This means that the shark is descending at a rate of 44 meters every minute.