Let's break down the problem step by step based on the given information and select the correct answers from the provided options:
Firstly, we need to understand the denominators in each of the functions.
1. For Practice Ride 1, the function is [tex]\( a(x) = \frac{5}{x} \)[/tex]. Here, [tex]\( x \)[/tex] represents Gilbert’s speed during his first practice ride.
2. For Practice Ride 2, the function is [tex]\( b(x) = \frac{9}{x + 2} \)[/tex]. Here, the denominator [tex]\( x + 2 \)[/tex] indicates that his speed during the second practice ride is [tex]\( x + 2 \)[/tex] miles per hour. Therefore, the denominator represents Gilbert's speed during the second practice ride.
Now, to find the total amount of time Gilbert spent on both practice rides, we need to add the times for each ride. The total amount of time function is obtained by:
[tex]\[ \text{Total Time}(x) = a(x) + b(x) = \frac{5}{x} + \frac{9}{x + 2} \][/tex]
This function, [tex]\( \frac{5}{x} + \frac{9}{x + 2} \)[/tex], models the total amount of time Gilbert spent on both practice rides.
Next, let's plug these findings into the given text with drop-down options:
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Gilbert is training for a bike race. As part of his training, he does practice rides on portions of the actual race course. Gilbert's first practice ride covers 5 miles of the course, and his second practice ride covers 9 miles of the course. Between these practice rides, he increases his average speed by 2 miles/hour.
These functions model the time it took Gilbert to do each practice ride, where [tex]\( x \)[/tex] is his speed during the first practice ride.
\begin{tabular}{|c|c|}
\hline Practice Ride 1 & Practice Ride 2 \\
\hline[tex]$a(x)=\frac{5}{x}$[/tex] & [tex]$b(x)=\frac{9}{x+2}$[/tex] \\
\hline
\end{tabular}
The denominator of the function that models practice ride 2 represents the speed during the second practice ride. To find a function that models the total amount of time Gilbert spent doing practice rides on the race course, add the functions.
