MULTI-STEP PROBLEM Two friends are 60 miles apart. They...

MULTI-STEP PROBLEM Two friends are 60 miles apart. They decide to ride their bicycles to meet each other. Sally starts from the college and heads east riding at a rate of 21 miles per hour. At the same time Teresa starts from the river and heads west, riding at a rate of 15 miles per hour. a. Draw a diagram showing the college and the river 60 miles apart and the two cyclists riding toward each other. b. How far does each cyclist ride in $t$ hours? c. When the two cyclists meet, what must be true about the distances they have ridden? Write and solve an equation to find when they meet. d. CHoosing A DATA DISPLAY Would a park that is 26 miles west of the river be a good meeting place? Explain. Use a diagram, a table, or a graph to support your answer.

MULTI-STEP PROBLEM Two friends are 60 miles apart. They decide to ride their bicycles to meet each other. Sally starts from the college and heads east riding at a rate of 21 miles per hour. At the same time Teresa starts from the river and heads west, riding at a rate of 15 miles per hour.
a. Draw a diagram showing the college and the river 60 miles apart and the two cyclists riding toward each other.
b. How far does each cyclist ride in $t$ hours?
c. When the two cyclists meet, what must be true about the distances they have ridden? Write and solve an equation to find when they meet.
d. CHoosing A DATA DISPLAY Would a park that is 26 miles west of the river be a good meeting place? Explain. Use a diagram, a table, or a graph to support your answer.

Key Concepts

Multiple Representations

Multiple representations such as diagrams, tables, or graphs are valuable tools for visualizing and organizing information in multi-step problems. They help provide a clear picture of the relationships between different variables and support the logical progression from problem setup to solution. This approach not only aids in understanding the problem but also in verifying the accuracy of the computed results.

Problem Modeling

Problem modeling involves translating a real-world scenario into mathematical expressions and equations. This process includes identifying the known quantities, assigning variables to the unknowns, and using appropriate formulas and relationships—such as the uniform motion formula—to depict the situation accurately. Creating a model is a crucial step in solving word problems efficiently.

Uniform Motion

Uniform motion describes movement at a constant speed over time, which means the distance traveled can be calculated as the product of speed and time. This is a fundamental concept in algebra and physics that allows one to set up linear equations relating distance, speed, and time, making it easier to solve motion problems under constant velocity conditions.

Relative Speed

Relative speed is the combined rate at which two objects approach or separate from each other when they are moving along the same line. In problems where two entities move towards each other, their relative speed is the sum of their individual speeds, and this combined speed can be used to establish an equation that determines the time until they meet. This concept simplifies the analysis of motion problems involving multiple bodies.

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