a man swims at a speed of 50 m/min in stil water, swims 100m against the current and 100m with the current. if the difference between the two times is 3 m 45 s, find the speed of the current
To solve this problem, we can use the formula:
Speed = Distance / Time
Let's first calculate the time it took to swim against the current:
Distance = 100m
Speed of the man = 50m/min (in still water)
Using the formula, we can find the time taken to swim against the current:
Time against current = Distance / Speed against current
= 100m / (50m/min - c)
Here, 'c' represents the speed of the current.
Next, let's calculate the time it took to swim with the current:
Distance = 100m
Speed of the man = 50m/min (in still water)
Using the formula, we can find the time taken to swim with the current:
Time with current = Distance / Speed with current
= 100m / (50m/min + c)
According to the problem statement, the difference between the two times is 3 minutes and 45 seconds:
Time with current - Time against current = 3 minutes 45 seconds
Converting 3 minutes 45 seconds to minutes, we have:
3 minutes 45 seconds = 3 minutes + (45 seconds / 60)
= 3 minutes + 0.75 minutes
= 3.75 minutes
Now, let's substitute the values in the equation and solve for 'c':
(100m / (50m/min + c)) - (100m / (50m/min - c)) = 3.75 minutes
We can simplify this equation further by cross-multiplication:
100m(50m/min - c) - 100m(50m/min + c) = 3.75 minutes[(50m/min + c)(50m/min - c)]
Expanding and simplifying the equation:
5000m - 100mc - 5000m - 100mc = 3.75 minutes(2500m^2 - c^2)
Simplifying further:
-200mc + 200mc = 3.75 minutes(2500m^2 - c^2)
The terms -200mc and +200mc cancel out, leaving us:
0 = 3.75 minutes(2500m^2 - c^2)
Simplifying, we have:
0 = 9375m^2 - 3.75c^2
Rearranging the equation:
3.75c^2 = 9375m^2
Dividing both sides by 3.75:
c^2 = 2500m^2
Taking the square root of both sides:
c = 50m
Therefore, the speed of the current is 50m/min.
To find the speed of the current, we can set up the following equation:
Speed of the man in still water = Speed of the man against the current - Speed of the current
Let's assume the speed of the current is denoted as "c" m/min.
Based on the given information, we can determine the time it takes for the man to swim 100m against the current and 100m with the current.
Time to swim against the current: Distance / (Speed in still water - Speed of current)
Time to swim with the current: Distance / (Speed in still water + Speed of current)
Given that the time difference between swimming against and with the current is 3 minutes and 45 seconds, we can convert it to seconds: 3 minutes * 60 seconds + 45 seconds = 225 seconds.
Now we have all the information to set up the equation:
100 / (50 - c) - 100 / (50 + c) = 225
Let's solve this equation to find the value of "c".
Multiplying both sides of the equation by (50 - c) * (50 + c):
100 * (50 + c) - 100 * (50 - c) = 225 * (50 - c) * (50 + c)
Expanding and simplifying the equation further:
5000 + 100c - 5000 + 100c = 225 * (2500 - c^2)
Combining like terms:
200c = 225 * (2500 - c^2)
Dividing both sides of the equation by 225:
c^2 - 2500/225 = 0
Simplifying the equation:
c^2 - 11.11 = 0
Rearranging the equation:
c^2 = 11.11
Taking the square root of both sides:
c = √11.11
Approximating the value of c:
c ≈ 3.33
Therefore, the speed of the current is 3.33 m/min.