Question

michael is on a train trip to treevale city. the train is traveling at a constant rate. after 5 hours on the train, there are 625 kilometers remaining to treevale city. after 8 hours, there are 484 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 michael from treevale city at the beginning of the trip?
\\(\square\\) kilometers

Explanation:

Part (a)

Step1: Find the rate of change

We have two points: at \( t_1 = 5 \) hours, distance remaining \( d_1 = 625 \) km; at \( t_2 = 8 \) hours, distance remaining \( d_2 = 484 \) km. The change in time \( \Delta t=t_2 - t_1=8 - 5 = 3 \) hours. The change in distance \( \Delta d=d_2 - d_1=484 - 625=- 141 \) km. The rate of change of distance with respect to time is \( \frac{\Delta d}{\Delta t}=\frac{- 141}{3}=- 47 \) km/h. The negative sign indicates that as time increases, the distance remaining decreases. So the rate is 47 kilometers per hour (the absolute value since we are talking about the rate of decrease of remaining distance).

Step2: Choose the correct statement

Since the rate is negative (distance remaining decreases as time increases) and the magnitude of the rate is 47 km/h, the correct statement is "As time increases, the distance remaining to Treevale City decreases. The distance remaining to Treevale City decreases at a rate of 47 kilometers per hour."

Step1: Determine the speed of the train

From part (a), we know the train is decreasing the remaining distance at a rate of 47 km/h. This means the train's speed is 47 km/h (since it's covering distance at that rate).

Step2: Find the distance traveled in 5 hours

Using the formula \( \text{distance}=\text{speed}\times\text{time} \), in 5 hours, the train travels \( 47\times5 = 235 \) km.

Step3: Find the initial distance

At 5 hours, the remaining distance is 625 km. So the initial distance (at the start of the trip) is the distance traveled in 5 hours plus the remaining distance at 5 hours. So \( 235+625 = 860 \) km.

Answer:

(for part a rate):
47

Part (b)