- Identify the points for the interval \([0.75, 1.00]\) hours as \((0.75, 3.50)\) and \((1.00, 4.75)\).
- Apply the average rate of change formula: \(\frac{f(t_2) - f(t_1)}{t_2 - t_1}\).
- Calculate the differences: \(1.25 \text{ miles}\) over \(0.25 \text{ hours}\).
- The average rate of change is \(\frac{1.25}{0.25} = 5.00 \text{ miles per hour}\), which is option (a).
- Final Answer: \(\boxed{5.00}\)
Explanation
- Identifying Data and Formula
The problem asks for the average rate of change of the distance traveled by a runner over a specific time interval. The average rate of change of a function \(f(t)\) over an interval \([t_1, t_2]\) is defined as the change in the distance divided by the change in time:
\[\text{Average Rate of Change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}\]
From the provided table, we can identify the coordinates \((t, d)\) at the given time points:
- For \(t_1 = 0.75\) hours, the distance is \(d_1 = 3.50\) miles.
- For \(t_2 = 1.00\) hours, the distance is \(d_2 = 4.75\) miles.
- Performing the Calculation
We substitute the values identified from the table into the average rate of change formula:
\[\text{Average Rate of Change} = \frac{4.75 - 3.50}{1.00 - 0.75}\]
First, calculate the change in distance (the numerator):
\[4.75 - 3.50 = 1.25 \text{ miles}\]
Next, calculate the change in time (the denominator):
\[1.00 - 0.75 = 0.25 \text{ hours}\]
Now, divide the change in distance by the change in time:
\[\frac{1.25}{0.25} = 5.00 \text{ miles per hour}\]
Comparing this result to the given options, we see that it corresponds to option (a).
Examples
In real-life scenarios, calculating the average rate of change is essential for understanding performance trends. For example, if you are driving a car and want to know your average speed between two points on a trip, you would divide the distance covered by the time taken. Similarly, in economics, a business might use this method to calculate the average growth rate of sales over a specific quarter, helping them identify whether their performance is improving or declining over time. This foundational skill helps in interpreting data and making informed decisions in various fields, from sports analytics to financial planning.