Question
Emma is on a cross-country road trip! She stops at a rest station to fill her gas tank and plans to drive for another 6 hours along the highway before making her next stop.
There is a linear relationship between the amount of gas left in her car's tank, in gallons, and the time since her last stop, in hours.
Time (hours) Amount of gas left (gallons)
1 9
1
2
2 7
3 4
1
2
4 2
Describe the rate of change for this relationship.
The amount of gas left
by
per hour.
There is a linear relationship between the amount of gas left in her car's tank, in gallons, and the time since her last stop, in hours.
Time (hours) Amount of gas left (gallons)
1 9
1
2
2 7
3 4
1
2
4 2
Describe the rate of change for this relationship.
The amount of gas left
by
per hour.
Answers
Answered by
GPT-4o mini
To find the rate of change of the amount of gas left in Emma's car tank with respect to time, we will first determine the relationship based on the data given.
We have the following points based on the time (in hours) and the amount of gas (in gallons) left:
- At 1 hour: 9 gallons
- At 2 hours: 7 gallons
- At 3 hours: 4 gallons
- At 4 hours: 2 gallons
The gas usage for each hour is calculated as follows:
- From hour 1 to hour 2: Gas decreases from 9 gallons to 7 gallons.
Change in gas = 7 - 9 = -2 gallons; Change in time = 1 hour; Rate of change = -2 gallons/1 hour = -2 gallons/hour.
- From hour 2 to hour 3: Gas decreases from 7 gallons to 4 gallons.
Change in gas = 4 - 7 = -3 gallons; Change in time = 1 hour; Rate of change = -3 gallons/1 hour = -3 gallons/hour.
- From hour 3 to hour 4: Gas decreases from 4 gallons to 2 gallons.
Change in gas = 2 - 4 = -2 gallons; Change in time = 1 hour; Rate of change = -2 gallons/1 hour = -2 gallons/hour.
It looks like the relationship is approximately as follows:
- From hours 1 to 2, the car consumes gas at the rate of -2 gallons/hour.
- From hours 2 to 3, the car consumes gas at the rate of -3 gallons/hour.
- From hours 3 to 4, the car again consumes gas at the rate of -2 gallons/hour.
To find the average rate of change over the entire 4 hours:
- Initial gas at hour 0: Unknown, but we can calculate using the first known variable at hour 1.
Assumed starting amount: If we backtrack using the data, we can assume it started at around 10 gallons, as she said she had 9 gallons after the first hour.
Final amount after 4 hours: 2 gallons left.
**Average rate of change over 4 hours**:
(Initial amount - Final amount) / (Total time)
(10 gallons - 2 gallons) / 4 hours = 8 gallons / 4 hours = -2 gallons/hour
Thus, the overall rate of change of gas is approximately **–2 gallons per hour**.
The conclusion is:
The amount of gas left decreases by approximately 2 gallons per hour.
We have the following points based on the time (in hours) and the amount of gas (in gallons) left:
- At 1 hour: 9 gallons
- At 2 hours: 7 gallons
- At 3 hours: 4 gallons
- At 4 hours: 2 gallons
The gas usage for each hour is calculated as follows:
- From hour 1 to hour 2: Gas decreases from 9 gallons to 7 gallons.
Change in gas = 7 - 9 = -2 gallons; Change in time = 1 hour; Rate of change = -2 gallons/1 hour = -2 gallons/hour.
- From hour 2 to hour 3: Gas decreases from 7 gallons to 4 gallons.
Change in gas = 4 - 7 = -3 gallons; Change in time = 1 hour; Rate of change = -3 gallons/1 hour = -3 gallons/hour.
- From hour 3 to hour 4: Gas decreases from 4 gallons to 2 gallons.
Change in gas = 2 - 4 = -2 gallons; Change in time = 1 hour; Rate of change = -2 gallons/1 hour = -2 gallons/hour.
It looks like the relationship is approximately as follows:
- From hours 1 to 2, the car consumes gas at the rate of -2 gallons/hour.
- From hours 2 to 3, the car consumes gas at the rate of -3 gallons/hour.
- From hours 3 to 4, the car again consumes gas at the rate of -2 gallons/hour.
To find the average rate of change over the entire 4 hours:
- Initial gas at hour 0: Unknown, but we can calculate using the first known variable at hour 1.
Assumed starting amount: If we backtrack using the data, we can assume it started at around 10 gallons, as she said she had 9 gallons after the first hour.
Final amount after 4 hours: 2 gallons left.
**Average rate of change over 4 hours**:
(Initial amount - Final amount) / (Total time)
(10 gallons - 2 gallons) / 4 hours = 8 gallons / 4 hours = -2 gallons/hour
Thus, the overall rate of change of gas is approximately **–2 gallons per hour**.
The conclusion is:
The amount of gas left decreases by approximately 2 gallons per hour.