Answer:
Let's denote the speed of the boat in still water as "B" (in miles per hour) and the speed of the current as "C" (in miles per hour).
When the boat is traveling upstream (against the current), its effective speed is reduced by the speed of the current. So, the relative speed going upstream is (B - C) miles per hour.
When the boat is traveling downstream (with the current), its effective speed is increased by the speed of the current. So, the relative speed going downstream is (B + C) miles per hour.
We are given two pieces of information:
1. The boat travels 6 miles upstream in 6 hours. This can be expressed as:
Distance = Speed × Time
6 miles = (B - C) miles per hour × 6 hours
2. Going downstream, it can travel 42 miles in the same amount of time (6 hours):
42 miles = (B + C) miles per hour × 6 hours
Now, let's solve these two equations simultaneously:
From the first equation:
6 = 6(B - C)
Divide both sides by 6:
1 = B - C
From the second equation:
42 = 6(B + C)
Divide both sides by 6:
7 = B + C
Now, we have a system of two equations:
1. B - C = 1
2. B + C = 7
We can solve this system of equations by adding the two equations together, which eliminates the "C" variable:
(B - C) + (B + C) = 1 + 7
This simplifies to:
2B = 8
Now, divide both sides by 2 to solve for B (the speed of the boat in still water):
B = 4 miles per hour
Now that we have the speed of the boat in still water (B), we can find the speed of the current (C) by using the first equation:
B - C = 1
4 - C = 1
Subtract 1 from both sides to solve for C:
C = 4 - 1
C = 3 miles per hour
So, the speed of the boat in still water is 4 miles per hour, and the speed of the current is 3 miles per hour.