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 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 0,60, B at ordered pair 1, 100, A at ordered pair 2, 140, and the ordered pair 3, 180.
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:
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, we can look at the similar triangles formed by the lines connecting these points to the origin (0,0). The triangles formed by points A, B, and the origin and points A, C, and the origin are similar because they have the same angles. This means that the ratios of the vertical distances to the horizontal distances for these triangles will be the same, which is the definition of slope. Therefore, the slopes between points A and B and points A and C will be equal.
Part B:
The initial value of the graph is 60, as this is the distance from the ocean surface at the start (x=0). The slope of the graph can be calculated by taking any two points, such as points A and B, and using the formula for slope: (change in y) / (change in x). In this case, the slope is (100-60) / (1-0) = 40. This slope represents the rate at which the shark is descending in meters per minute. So, the initial value of 60 represents the starting depth of the shark, and the slope of 40 represents the rate at which the shark is descending.