Question

the table shows how the time it takes a train to travel between two cities depends on its average speed.

average speed, x (mph)	time, y (hr)
40	4
50	3.2
64	2.5

which rational function models the time, y, in hours, that it takes the train to travel between the two cities at an average speed of x miles per hour?
what does the number 160 represent in the rational function that models the situation?

Explanation:

Step1: Recall the formula for distance

The formula relating speed (\(x\)), time (\(y\)), and distance (\(d\)) is \(d = x\times y\) (distance = speed × time). We can find the distance between the two cities using any of the given speed - time pairs. Let's use the first pair: \(x = 32\) mph and \(y = 5\) hours. Then \(d=32\times5 = 160\) miles. We can check with other pairs: for \(x = 40\) mph and \(y = 4\) hours, \(d = 40\times4=160\) miles; for \(x = 50\) mph and \(y = 3.2\) hours, \(d=50\times3.2 = 160\) miles; for \(x = 64\) mph and \(y = 2.5\) hours, \(d = 64\times2.5=160\) miles.

Step2: Find the rational function for time

Since \(d=x\times y\) and \(d = 160\), we can solve for \(y\) in terms of \(x\). Rearranging the formula \(d=x\times y\) gives \(y=\frac{d}{x}\). Substituting \(d = 160\) into the formula, we get the rational function \(y=\frac{160}{x}\).

Step3: Interpret the number 160

From the formula \(d=x\times y\) (distance = speed × time), we found that the distance between the two cities is constant (160 miles) for all the given speed - time pairs. So the number 160 represents the distance (in miles) between the two cities.

Answer:

(for the first question: rational function):
The rational function that models the time \(y\) (in hours) that it takes the train to travel between the two cities at an average speed of \(x\) miles per hour is \(y=\frac{160}{x}\).