This interactive tool helps you better understand the link between the visual and graphical approaches to time, rate, and distance problems and their algebraic solutions.
Tutorial Objectives
- Learn how to use the interactive simulation to approximate a solution to the problem.
- Find a precise algebraic solution to the problem with the help of graphical insights.
Problem 1
At 11:00 AM, car (1) leaves city "A" at a constant rate of \( 60\text{ mi/hr} \) toward city "B". At the same time, a second car (2) leaves city "B" toward city "A" at a constant rate of \( 50\text{ mi/hr} \). The distance between cities A and B is 220 miles, and these cities are connected by a highway used by the two cars.
Question: At what time will the two cars cross each other?
Interactive Graphical Simulation
Use the controls below to run the simulation, step through the motion, or pause to observe the distance-time graph intersection.
Instructions:
- Click "Continuous / Start" to watch the two cars travel along the highway. On the right side of the canvas, two graphs show the distances of car (1) (in blue) and car (2) (in red) from city A. The blue graph starts at distance zero, while the red graph starts at 220 miles. When the cars meet, the two graphs intersect.
- Click "Pause" to suspend the animation at any time.
- Click "Reset All" to return the cars to their initial positions at 11:00 AM (\( t = 0 \)).
- Click "By Step" repeatedly to advance time incrementally until the two graphs intersect and read the time approximation.
Algebraic Solution
Let \( t \) represent the time in hours elapsed since 11:00 AM (\( t = 0 \)):
- The distance \( d_1 \) between city "A" and car (1) changes with time as: \[ d_1 = 60t \]
- The distance \( d_2 \) between city "A" and car (2) changes with time as: \[ d_2 = 220 - 50t \]
When the cars cross each other, their distances from city A are equal (\( d_1 = d_2 \)), corresponding to the intersection of the blue and red graphs:
\[ 60t = 220 - 50t \] \[ 110t = 220 \] \[ t = 2 \text{ hours} \]Therefore, the two cars cross each other at:
\[ 11:00\text{ AM} + 2\text{ hours} = 13:00\text{ (1:00 PM)} \]Compare this exact algebraic result to the approximation obtained from the interactive graphical simulation above.