Question

1. bloodhound ssc - predicted distance time graph what is the average rate of change from 0 to 40 seconds? a) 0.15 mps. b) 3 mps. c) 0.5 mps. d) 6 mps.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( y = f(x) \) over the interval \([x_1, x_2]\) is given by \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). In the context of distance - time graph, the average rate of change is the average speed, and it is calculated as \(\frac{\text{Change in distance}}{\text{Change in time}}\).

Step2: Identify the values from the graph

From the graph, at \( t = 0 \) seconds, the distance \( d_1=0\) miles (since the graph starts at the origin \((0,0)\)). At \( t = 40 \) seconds, the distance \( d_2 = 6\) miles (from the point \((40,6)\) on the graph).

Step3: Calculate the average rate of change

The change in distance \(\Delta d=d_2 - d_1=6 - 0=6\) miles. The change in time \(\Delta t=t_2 - t_1=40 - 0 = 40\) seconds.
The average rate of change (average speed) \(r=\frac{\Delta d}{\Delta t}=\frac{6}{40}=0.15\) mps? Wait, no, wait. Wait, maybe I misread the graph. Wait, let's check again. Wait, the y - axis is distance in miles? Wait, no, maybe the units are different? Wait, no, the graph: at \( t = 40 \) seconds, the distance is 6? Wait, no, maybe the y - axis is in miles? Wait, no, let's recalculate. Wait, the formula for average rate of change is \(\frac{\text{Final distance}-\text{Initial distance}}{\text{Final time}-\text{Initial time}}\).
Initial time \(t_1 = 0\) s, initial distance \(d_1=0\) m (assuming the y - axis is in miles? Wait, no, maybe the y - axis is in miles? Wait, no, the problem says "distance (miles)"? Wait, the graph: the vertical axis is "Distance (miles)"? Wait, at \( t = 40 \) seconds, the distance is 6 miles? Wait, no, let's do the calculation again.
\(r=\frac{d(40)-d(0)}{40 - 0}\). From the graph, \(d(0) = 0\), \(d(40)=6\). So \(r=\frac{6 - 0}{40-0}=\frac{6}{40}=0.15\) mps? But wait, that's option a. Wait, but let's check the options. Option a is 0.15 mps, option c is 0.5, option b is 3, option d is 6. Wait, maybe I made a mistake in the distance. Wait, maybe the y - axis is in kilometers? No, the problem says "distance (miles)"? Wait, no, maybe the graph's y - axis is in miles, but the time is in seconds. Wait, let's recalculate: \(\frac{6}{40}=0.15\) mps. So the average rate of change is \(\frac{6}{40}=0.15\) mps.

Answer:

a) 0.15 mps