To find the speed of the current, let's define [tex]\(x\)[/tex] as the speed of the current in miles per hour.
1. Speed of the boat with the current:
- When the boat is traveling with the current, the effective speed is the boat's speed in still water plus the speed of the current.
- Effective speed with current = [tex]\(10 + x\)[/tex] miles per hour.
2. Speed of the boat against the current:
- When the boat is traveling against the current, the effective speed is the boat's speed in still water minus the speed of the current.
- Effective speed against current = [tex]\(10 - x\)[/tex] miles per hour.
3. Using the relationship [tex]\(t = \frac{d}{r}\)[/tex]:
- Time to travel 6 miles with the current:
[tex]\[
t_{\text{with}} = \frac{6}{10 + x}
\][/tex]
- Time to travel 4 miles against the current:
[tex]\[
t_{\text{against}} = \frac{4}{10 - x}
\][/tex]
4. Since the times are equal:
[tex]\[
\frac{6}{10 + x} = \frac{4}{10 - x}
\][/tex]
5. Solve the rational equation:
- Cross-multiplying to get rid of the denominators:
[tex]\[
6(10 - x) = 4(10 + x)
\][/tex]
- Expanding both sides:
[tex]\[
60 - 6x = 40 + 4x
\][/tex]
- Combining like terms:
[tex]\[
60 - 40 = 4x + 6x
\][/tex]
[tex]\[
20 = 10x
\][/tex]
- Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{20}{10} = 2
\][/tex]
Therefore, the speed of the current is [tex]\(2\)[/tex] miles per hour.
In summary:
- The correct answer is [tex]\(x = 2\)[/tex].
