Question

dan is on a train trip to treevale city. the train is traveling at a constant rate. after 6 hours on the train, there are 576 kilometers remaining to treevale city. after 11 hours, there are 331 kilometers remaining to treevale city.
(a) choose the statement that best describes how the time and the distance remaining to treevale city are related. then fill in the blank.
○ as time increases, the distance remaining to treevale city decreases.
the distance remaining to treevale city decreases at a rate of \\(\square\\) kilometers per hour.
○ as time increases, the distance remaining to treevale city increases.
the distance remaining to treevale city increases at a rate of \\(\square\\) kilometers per hour.
(b) how far was dan from treevale city at the beginning of the trip?
\\(\square\\) kilometers

Explanation:

Part (a)

Step1: Analyze the relationship

As time on the train increases, the distance remaining to the city should decrease (since the train is moving towards the city). So we choose the first option.

Step2: Calculate the rate of decrease

The time difference is \(11 - 6 = 5\) hours. The distance difference is \(576 - 331 = 245\) kilometers. The rate of decrease is \(\frac{245}{5}=49\) kilometers per hour.

Step1: Use the rate to find initial distance

We know after 6 hours, distance remaining is 576 km. The train travels at 49 km/h. In 6 hours, it travels \(49\times6 = 294\) km.

Step2: Find initial distance

Initial distance = distance traveled in 6 hours + distance remaining after 6 hours. So \(294 + 576 = 870\) km.

Answer:

As time increases, the distance remaining to Treevale City decreases.
The distance remaining to Treevale City decreases at a rate of \(\boldsymbol{49}\) kilometers per hour.

Part (b)