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A boat can travel at an average speed of 10 miles per hour in still water. Traveling with the current, it can travel 6 miles in the same amount of time it takes to travel 4 miles upstream.

Use the relationship [tex]t=\frac{d}{r}[/tex] to construct a rational equation that can be solved for [tex]x[/tex] to find the speed of the current.



Answer :

GinnyAnswer
To solve for the speed of the current, let's go through the steps systematically.

1. Define the Variables:
- Speed of the boat in still water: [tex]\( 10 \)[/tex] miles per hour
- Distance traveled downstream: [tex]\( 6 \)[/tex] miles
- Distance traveled upstream: [tex]\( 4 \)[/tex] miles
- Speed of the current: [tex]\( x \)[/tex] miles per hour

2. Formulate the Time Equations:
- When traveling downstream, the effective speed of the boat is the speed of the boat in still water plus the speed of the current: [tex]\( 10 + x \)[/tex].
- Time taken to travel downstream [tex]\( t_1 = \frac{6}{10 + x} \)[/tex]
- When traveling upstream, the effective speed of the boat is the speed of the boat in still water minus the speed of the current: [tex]\( 10 - x \)[/tex].
- Time taken to travel upstream [tex]\( t_2 = \frac{4}{10 - x} \)[/tex]

3. Set Up the Rational Equation:
Since the times taken to travel the given distances downstream and upstream are the same, we equate the two times:
[tex]\[ \frac{6}{10 + x} = \frac{4}{10 - x} \][/tex]

4. Solve for [tex]\( x \)[/tex]:
- Cross-multiplying the equation gives:
[tex]\[ 6 \cdot (10 - x) = 4 \cdot (10 + x) \][/tex]
- Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 60 - 6x = 40 + 4x \][/tex]
[tex]\[ 60 - 40 = 4x + 6x \][/tex]
[tex]\[ 20 = 10x \][/tex]
[tex]\[ x = 2 \][/tex]

Therefore, the speed of the current is [tex]\( \boxed{2} \)[/tex] miles per hour.

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