Mai is driving a racecar. The table below gives the distance [tex]\( D(t) \)[/tex] (in meters) she has driven at a few times [tex]\( t \)[/tex] (in seconds) after she starts.

[tex]\[
\begin{tabular}{|c|c|}
\hline
Time \( t \) (seconds) & Distance \( D(t) \) (meters) \\
\hline
0 & 0 \\
\hline
2 & 78.6 \\
\hline
5 & 151.5 \\
\hline
7 & 205.1 \\
\hline
9 & 255.9 \\
\hline
\end{tabular}
\][/tex]

(a) Find the average rate of change for the distance driven from 0 seconds to 2 seconds.

39.3 meters per second

(b) Find the average rate of change for the distance driven from 5 seconds to 9 seconds.

[tex]\(\square\)[/tex] meters per second



Answer :

To solve part (b) of the problem, we need to find the average rate of change for the distance driven from 5 seconds to 9 seconds.

Step 1: Identify the given data points for time and distance.
- At [tex]\( t = 5 \)[/tex] seconds, the distance [tex]\( D(t) = 151.5 \)[/tex] meters.
- At [tex]\( t = 9 \)[/tex] seconds, the distance [tex]\( D(t) = 255.9 \)[/tex] meters.

Step 2: Use the formula for average rate of change. The average rate of change of a function [tex]\( D(t) \)[/tex] over an interval from [tex]\( t = a \)[/tex] to [tex]\( t = b \)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{D(b) - D(a)}{b - a} \][/tex]
In this problem, [tex]\( a = 5 \)[/tex] and [tex]\( b = 9 \)[/tex].

Step 3: Substitute the given values into the formula.

[tex]\[ \text{Average Rate of Change} = \frac{D(9) - D(5)}{9 - 5} = \frac{255.9 \text{ meters} - 151.5 \text{ meters}}{9 \text{ seconds} - 5 \text{ seconds}} \][/tex]

Step 4: Calculate the differences.

[tex]\[ \text{Distance Difference} = 255.9 \text{ meters} - 151.5 \text{ meters} = 104.4 \text{ meters} \][/tex]
[tex]\[ \text{Time Difference} = 9 \text{ seconds} - 5 \text{ seconds} = 4 \text{ seconds} \][/tex]

Step 5: Compute the average rate of change.

[tex]\[ \text{Average Rate of Change} = \frac{104.4 \text{ meters}}{4 \text{ seconds}} = 26.1 \text{ meters per second} \][/tex]

Thus, the average rate of change for the distance driven from 5 seconds to 9 seconds is [tex]\(\boxed{26.1}\)[/tex] meters per second.

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