The Gulf Stream is a warm ocean current that extends from...

The Gulf Stream is a warm ocean current that extends from the eastern side of the Gulf of Mexico up through the Florida Straits and along the southeastern coast of the United States to Cape Hatteras, North Carolina. A boat travels with the current $100 \mathrm{mi}$ from Miami, Florida, to Free port, Bahamas, in 2.5 hr. The return trip against the same current takes $3 \frac{1}{3}$ hr. Find the speed of the boat in still water and the speed of the current.

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Key Concepts

Systems of Linear Equations

When two different scenarios of a related problem (like upstream and downstream travel) are provided, setting up a system of equations enables the simultaneous solving for unknown quantities. This method involves creating equations based on the different conditions and solving them concurrently to find the values of the unknown variables.

Relative Motion

In problems involving movement in a medium (like a boat in a current), the effective speed is determined by combining the object's own speed with the speed of the medium. When the object moves with the current, these speeds add together; when the object moves against the current, the current's speed subtracts from the object's speed in still water.

Rate-Time-Distance Relationship

A fundamental principle in motion problems is the relationship between distance, speed, and time, where distance is the product of speed and time. This relationship is used to set up equations for different scenarios (such as moving with and against a current) to analyze the dynamics of motion.

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