Distance between Cars Cars A and B are both traveling in the...

Distance between Cars Cars A and B are both traveling in the same direction. Their distances in miles north of St. Louis after $x$ hours are computed by the functions $f_{A}$ and $f_{B}$ respectively. The graphs of $f_{A}$ and $f_{B}$ are shown in the figure for $0 \leq x \leq 10$ (a) Which car is traveling faster? Explain. (b) How many hours elapse before the two cars are the same distance from St. Louis? How far are they from St. Louis when this occurs? (c) During what time interval is car $\mathbf{B}$ farther from $\mathbf{S t}$. Louis than car A?

Distance between Cars Cars A and B are both traveling in the same direction. Their distances in miles north of St. Louis after $x$ hours are computed by the functions $f_{A}$ and $f_{B}$ respectively. The graphs of $f_{A}$ and $f_{B}$ are shown in the figure for $0 \leq x \leq 10$
(a) Which car is traveling faster? Explain.
(b) How many hours elapse before the two cars are the same distance from St. Louis? How far are they from St. Louis when this occurs?
(c) During what time interval is car $\mathbf{B}$ farther from $\mathbf{S t}$. Louis than car A?

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Key Concepts

Inequalities and Interval Analysis

Analyzing inequalities in the context of distance-time graphs involves determining the time intervals during which one object's distance exceeds the other's. This helps in identifying periods when one object is ahead of the other, such as when a car is farther from the starting point compared to another.

Slope as Rate of Change

The slope of a line on a distance-time graph represents the object's speed or rate of change in distance over time. A steeper slope means the object is traveling faster. In comparing the slopes of two lines, one can determine which object is moving at a higher speed.

Distance-Time Graphs

A distance-time graph visually represents how far an object has traveled over time. In these graphs, the horizontal axis usually represents time while the vertical axis represents distance. This type of graph helps in analyzing motion by showing positions at various time points, which is critical for understanding problems involving moving objects.

Intersection of Graphs

An intersection point on two distance-time graphs indicates the moment when both objects have traveled the same distance from a common starting point. Finding this intersection helps in determining when the two moving objects are side-by-side or at the same position along the path.

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