To solve this problem, we'll derive a rational equation to find the speed of the current. Given that:
- The speed of the boat in still water is \( 10 \) miles per hour.
- The distance traveled downstream and upstream is \( 6 \) miles.
- The times taken for both journeys (downstream and upstream) are equal.
Let's denote the speed of the current as \( x \) miles per hour.
1. Downstream journey: The effective speed of the boat travelling downstream (with the current) is \( 10 + x \) miles per hour.
2. Upstream journey: The effective speed of the boat travelling upstream (against the current) is \( 10 - x \) miles per hour.
3. The time to travel a given distance \( d \) at speed \( r \) is given by: \( t = \frac{d}{r} \).
Given that the times for downstream and upstream journeys are equal, we can set their time equations equal to each other:
[tex]\[ \text{Time downstream} = \text{Time upstream} \][/tex]
Expressing these times using the given distance and the boat's effective speeds:
[tex]\[ \frac{6}{10 + x} = \frac{6}{10 - x} \][/tex]
To construct and solve the rational equation, follow these steps:
1. Set up the equation:
[tex]\[ \frac{6}{10 + x} = \frac{6}{10 - x} \][/tex]
2. Eliminate the fractions by multiplying both sides by the denominators (first multiply both sides by \((10 + x)(10 - x)\)):
[tex]\[ 6(10 - x) = 6(10 + x) \][/tex]
3. Simplify the equation:
[tex]\[ 60 - 6x = 60 + 6x \][/tex]
4. Combine like terms:
[tex]\[ 60 - 60 = 6x + 6x \][/tex]
[tex]\[ 0 = 12x \][/tex]
5. Solve for \( x \):
[tex]\[ x = 0 \][/tex]
Thus, the speed of the current \( x \) is 0 miles per hour.
So the rational equation that can be used to find the speed of the current is:
[tex]\[ \frac{6}{10 + x} = \frac{6}{10 - x} \][/tex]
