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Translate to a system of equations and then solve: Mitchell left Detroit on the interstate driving south towards Orlando at a speed of 60 miles per hour. Clark left Detroit 1 hour later traveling at a speed of 75 miles per hour, following the same route as Mitchell. How long will it take Clark to catch Mitchell?
It will take Clark 4 hours to catch Mitchell.
Translate to a system of equations and then solve: Charlie left his mother’s house traveling at an average speed of 36 miles per hour. His sister Sally left 15 minutes (1/4 hour) later traveling the same route at an average speed of 42 miles per hour. How long before Sally catches up to Charlie?
It will take Sally hours to catch up to Charlie.
Many real-world applications of uniform motion arise because of the effects of currents—of water or air—on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.
Let’s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.
[link] and [link] show how a river current affects the speed at which a boat is actually travelling. We’ll call the speed of the boat in still water b and the speed of the river current c .
In [link] the boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is b + c .
In [link] the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat’s actual speed is slower than its speed in still water. The actual speed of the boat is .
We’ll put some numbers to this situation in [link] .
Translate to a system of equations and then solve:
A river cruise ship sailed 60 miles downstream for 4 hours and then took 5 hours sailing upstream to return to the dock. Find the speed of the ship in still water and the speed of the river current.
Read the problem.
This is a uniform motion problem and a picture will help us visualize the situation.
| Identify what we are looking for. | We are looking for the speed of the ship
in still water and the speed of the current. |
| Name what we are looking for. | Let
the rate of the ship in still water.
the rate of the current |
| A chart will help us organize the information.
The ship goes downstream and then upstream. Going downstream, the current helps the ship; therefore, the ship’s actual rate is s + c . Going upstream, the current slows the ship; therefore, the actual rate is s − c . |
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| Downstream it takes 4 hours.
Upstream it takes 5 hours. Each way the distance is 60 miles. |
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Translate into a system of equations.
Since rate times time is distance, we can write the system of equations. |
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Solve the system of equations.
Distribute to put both equations in standard form, then solve by elimination. |
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| Multiply the top equation by 5 and the bottom equation by 4.
Add the equations, then solve for s . |
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| Substitute s = 13.5 into one of the original equations. |
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Check the answer in the problem.
The downstream rate would be 13.5 + 1.5 = 15 mph. In 4 hours the ship would travel 15 · 4 = 60 miles. The upstream rate would be 13.5 − 1.5 = 12 mph. In 5 hours the ship would travel 12 · 5 = 60 miles. |
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| Answer the question. | The rate of the ship is 13.5 mph and
the rate of the current is 1.5 mph. |
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