A climber is on a hike. After 2 hours he is at an altitude of 400 feet. After 6 hours, he is at an altitude of 800 feet. What is the average rate of change?
Understand the Problem
The question is asking for the average rate of change in altitude of a climber over a specified time period. To solve it, we will calculate the difference in altitude and divide it by the time difference.
Answer
The average rate of change in altitude is $100$ feet per hour.
Answer for screen readers
The average rate of change in altitude is $100$ feet per hour.
Steps to Solve
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Determine the change in altitude To find the total change in altitude, subtract the initial altitude from the final altitude. The initial altitude at 2 hours is 400 feet and the altitude at 6 hours is 800 feet. So, the change in altitude is: $$ \text{Change in altitude} = 800 \text{ feet} - 400 \text{ feet} = 400 \text{ feet} $$
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Calculate the time interval Next, find the total time elapsed. The climber starts at 2 hours and ends at 6 hours. The time interval is: $$ \text{Time interval} = 6 \text{ hours} - 2 \text{ hours} = 4 \text{ hours} $$
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Calculate the average rate of change Now, use the changes calculated to find the average rate of change in altitude. This is done by dividing the change in altitude by the time interval: $$ \text{Average rate of change} = \frac{\text{Change in altitude}}{\text{Time interval}} = \frac{400 \text{ feet}}{4 \text{ hours}} = 100 \text{ feet per hour} $$
The average rate of change in altitude is $100$ feet per hour.
More Information
The average rate of change gives us an understanding of how quickly the climber is gaining altitude over a specific period. In this case, the climber is ascending at an average speed of 100 feet for each hour spent hiking.
Tips
- Confusing the time intervals; ensure to subtract the correct hours.
- Neglecting to perform the subtraction for either the altitude or time correctly. Double-check calculations to avoid errors.
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