What is the rate of change? Find the change in Marco’s distance each hour. What is the initial value? Find Marco’s distance from the Grand Canyon when he starts to drive.
Understand the Problem
The question involves determining the rate of change in Marco's distance from the Grand Canyon over time, as well as finding his initial distance from the Grand Canyon when he begins his journey. The problem indicates the distance he travels decreases over time, providing specific values to work with.
Answer
Rate of change: $-50$ miles per hour; Initial value: $400$ miles.
Answer for screen readers
Rate of change: $-50$ miles per hour
Initial value: $400$ miles
Steps to Solve
- Calculate the rate of change
Marco's distance from the Grand Canyon decreases by 150 miles every 3 hours. To find the rate of change per hour, divide the total distance by the time in hours: $$ \text{Rate of change} = \frac{-150 \text{ miles}}{3 \text{ hours}} = -50 \text{ miles per hour} $$
- Determine the initial value
After 4 hours of driving, Marco's distance from the Grand Canyon is 200 miles. We can use this information to find the initial distance before he started his journey. Since he drives for 4 hours at a rate of -50 miles per hour, we can calculate the initial distance ( y_0 ) using the equation: $$ y_0 = \text{Final distance} + (\text{Rate} \times \text{Time}) $$ Substituting the known values: $$ y_0 = 200 + (50 \times 4) = 200 + 200 = 400 \text{ miles} $$
Rate of change: $-50$ miles per hour
Initial value: $400$ miles
More Information
The rate of change of $-50$ miles per hour indicates that Marco is getting closer to the Grand Canyon at a steady speed. The initial distance of $400$ miles shows how far he was from the Grand Canyon when he started his journey.
Tips
- Misunderstanding the Rate of Change: Some might calculate the rate considering a different time frame. Always ensure you're using the correct time interval.
- Forgetting to Convert: If you're given mixed units, be sure to convert them before performing calculations.
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