Let's break down the problem step by step:
1. Understanding the practice rides:
- Practice Ride 1:
Gilbert's speed during the first practice ride is [tex]\( x \)[/tex] miles per hour. He covers 5 miles. Therefore, the time taken for the first practice ride is given by the function:
[tex]\[
a(x) = \frac{5}{x}
\][/tex]
- Practice Ride 2:
Between the two practice rides, Gilbert increases his average speed by 2 miles per hour. Hence, his speed during the second practice ride is [tex]\( x + 2 \)[/tex] miles per hour. He covers 9 miles. Therefore, the time taken for the second practice ride is given by the function:
[tex]\[
b(x) = \frac{9}{x + 2}
\][/tex]
2. Combining the times:
To find the total time Gilbert spent doing both practice rides, we need to sum the individual times:
[tex]\[
\text{Total time} = a(x) + b(x) = \frac{5}{x} + \frac{9}{x + 2}
\][/tex]
3. Simplifying the total time function:
We can further simplify this expression to ensure it has a common denominator:
[tex]\[
\frac{5}{x} + \frac{9}{x + 2} = \frac{5(x + 2) + 9x}{x(x + 2)} = \frac{5x + 10 + 9x}{x(x + 2)} = \frac{14x + 10}{x(x + 2)}
\][/tex]
This can also be written as:
[tex]\[
\frac{2(7x + 5)}{x(x + 2)}
\][/tex]
4. Conclusion:
The denominator of the function that models practice ride 2 represents the biking speed for the second practice ride, which is [tex]\( x + 2 \)[/tex] miles per hour.
5. Function that models the total amount of time:
- Unmodified total time function is:
[tex]\[
\frac{5}{x} + \frac{9}{x + 2}
\][/tex]
- Simplified total time function is:
[tex]\[
\frac{2(7x + 5)}{x(x + 2)}
\][/tex]
Therefore, the correct answer selections are:
1. The denominator of the function that models practice ride 2 represents:
- The biking speed for the second practice ride.
2. To find a function that models the total amount of time Gilbert spent doing practice rides on the race course:
- Add the functions.
