Question

part 1: connecting arithmetic sequences and direct variation
a truck driver is beginning her haul across the country. the warehouse is 15 miles from the highway. once she gets on the highway she maintains a constant rate for several hours. complete the table of values for the sequence below that relates hours on the highway and distance from the warehouse is miles.

hours driven (h)	distance from warehouse (d)
1	80
2	145
3
4
5
6

1. how much does the truckers distance from the warehouse increase every hour?
2. what is the rate of change for the sequence?
3. can we figure out the trucks speed from this information? if so, what is it?
4. how can we figure out how far away from the warehouse the trucker will be in the next hour?
5. if the truck continues traveling at the same rate for 2 more hours, how many more miles will the distance from the warehouse increase?
6. what if the trucker continues at the same rate for 3 more hours? 4 more hours?
7. in general, if the trucker continues at the same rate for h more hours, how could we calculate how much farther away she will be?

Explanation:

Step1: Find distance increase per hour

Subtract distance at hour 0 from hour 1. $80 - 15=65$ miles.

Step2: Define rate of change

The rate of change is the change in distance per unit - time. Since the distance increases by 65 miles every hour, the rate of change is 65 miles per hour.

Step3: Determine truck's speed

The truck's speed is equal to the rate of change of the distance - time relationship. So the speed is 65 miles per hour.

Step4: Calculate next - hour distance

Add the speed (65 miles per hour) to the current distance. For example, if at hour $n$ the distance is $D_n$, at hour $n + 1$ the distance $D_{n+1}=D_n+65$.

Step5: Calculate distance increase for 2 hours

Multiply the speed by the number of hours. $65\times2 = 130$ miles.

Step6: Calculate distance increase for 3 and 4 hours

For 3 hours: $65\times3=195$ miles. For 4 hours: $65\times4 = 260$ miles.

Step7: General formula for distance increase

Multiply the speed (65 miles per hour) by the number of additional hours $h$. The additional distance $d = 65h$.

Answer:

1. 65 miles
2. 65 miles per hour
3. Yes, 65 miles per hour
4. Add 65 miles to the current distance from the warehouse.
5. 130 miles
6. 195 miles for 3 more hours, 260 miles for 4 more hours
7. $65h$ miles