Question

the table shows y, the average speed of a cyclist in miles per hour, and x, the time in hours that it takes the cyclist to complete a bicycle tour. which rational function best models the data in the table?

time, x (hours)	average speed, y (miles per hour)
16	6
$10\frac{2}{3}$	9
18	$5\frac{1}{3}$

options: $y = \frac{96}{x}$, $y = \frac{2x}{3}$, $y = \frac{x}{96}$, $y = \frac{3}{2x}$

Explanation:

Step1: Recall the formula for distance

The formula relating speed (\(y\)), time (\(x\)), and distance (\(d\)) is \(d = y\times x\) (distance = speed × time). For a bicycle tour, the distance should be constant, so we can find the distance using the given \(x\) and \(y\) values.

Step2: Calculate the distance using the first pair of values

For \(x = 12\) hours and \(y = 8\) miles per hour, the distance \(d=12\times8 = 96\) miles.

Step3: Verify the distance with other pairs

• For \(x = 16\) and \(y = 6\): \(16\times6=96\) miles.
• For \(x = 10\frac{2}{3}=\frac{32}{3}\) and \(y = 9\): \(\frac{32}{3}\times9 = 32\times3=96\) miles.
• For \(x = 18\) and \(y = 5\frac{1}{3}=\frac{16}{3}\): \(18\times\frac{16}{3}=6\times16 = 96\) miles.

Since \(d = 96\) (constant), from \(d=y\times x\), we can solve for \(y\) to get \(y=\frac{d}{x}=\frac{96}{x}\).

Answer:

\(y = \frac{96}{x}\) (the first function option)