To solve the problem, we need to understand the relationship between the time [tex]\( t \)[/tex], the distance [tex]\( d \)[/tex], and the speed [tex]\( m \)[/tex]. The problem states that the time [tex]\( t \)[/tex] it takes to complete the distance varies inversely with the speed [tex]\( m \)[/tex]. This means that as the speed increases, the time decreases, and vice versa.
The inverse relationship between time and speed can be expressed mathematically as:
[tex]\[ t \propto \frac{1}{m} \][/tex]
This means:
[tex]\[ t = \frac{k}{m} \][/tex]
where [tex]\( k \)[/tex] is a constant.
Next, we need to determine the value of the constant [tex]\( k \)[/tex]. The problem specifies that the distance Kevin cycles is 18 miles. When dealing with distance, speed, and time, we have the relationship:
[tex]\[ \text{distance} = \text{speed} \times \text{time} \][/tex]
In this case:
[tex]\[ d = m \times t \][/tex]
We know the distance [tex]\( d \)[/tex] is 18 miles:
[tex]\[ 18 = m \times t \][/tex]
To find the equation that represents the time [tex]\( t \)[/tex], we can solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{18}{m} \][/tex]
Thus, the equation that best models the amount of time [tex]\( t \)[/tex] it takes Kevin to finish cycling if he rides at a speed of [tex]\( m \)[/tex] miles per hour is:
[tex]\[ t = \frac{18}{m} \][/tex]
Therefore, the correct choice is:
A. [tex]\( t=\frac{18}{m} \)[/tex]
