3.
(05.05 MC)
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)
I need help fast!
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 using similar triangles, we can use the concept of proportionality.
Similar triangles have corresponding angles that are equal and proportional sides. In this case, we can consider the two right triangles formed between point A and the x-axis at points B and C.
Let's label the vertical side of the triangle between A and B as "a" and the corresponding vertical side of the triangle between A and C as "b". Similarly, let's label the horizontal side of the triangle between A and B as "c" and the corresponding horizontal side of the triangle between A and C as "d".
Since the triangles are similar, we can set up a proportion with the sides of the triangles:
a/c = b/d
We know that the slope of a line is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points. In this case, we can consider the slope between points A and B as m1 and the slope between points A and C as m2.
Therefore, we can equate the slopes to the proportion of the sides of the triangles:
m1 = a/c
m2 = b/d
Since the triangles are similar and have equal corresponding angles, the ratio of the sides remains the same. Hence, the slopes m1 and m2 will be equal.
Part B:
The initial value of the graph represents the y-intercept, which is the value of y when x is zero. In this case, the initial value is 60 meters. It means that when the shark starts swimming, it is initially 60 meters below the surface of the ocean.
The slope of the graph represents the rate of change of distance with respect to time. In this case, the slope is 40 (the change in distance is 40 meters for each unit change in time). It means that the shark is swimming at a constant speed of 40 meters per minute, moving further away from the surface with each passing minute.