A man rowing upstream drops his hat at point M. Thirty minutes later at N, he notices its loss and rows back. He picks up his hat at P, 440 yards below M. Find the rate of the stream and the total elapsed time.
Well, well, well, looks like our adventurous rower got into a hat-dropping situation! Let's dive into the depths of this river puzzle, shall we?
First things first, let's define some variables to make it easier. Let's call the rate of the man's rowing "R" and the rate of the stream "S".
Now, when the man drops his hat at point M, it starts drifting downstream due to the stream. Let's calculate how far the hat has traveled during those 30 minutes. Since the speed of the stream is S and the hat has been drifting for 30 minutes (or a half hour), the hat has traveled 0.5*S miles downstream.
When the man realizes his hat is missing at point N, he decides to turn around and row back upstream to retrieve it. This time, he rows at a rate of R-S (since he's rowing against the current caused by the stream).
By the time the man reaches point P to pick up his hat, he has rowed a total distance of 440 yards. This means that his rowing speed of R-S for 30 minutes corresponds to 440 yards.
Now let's convert everything to the same units. Since we have yards and minutes, let's keep it consistent. 440 yards is the same as 0.25 miles (since 1 mile = 1760 yards).
So, we have the equation:
(R-S) * 0.5 = 0.25
Simplifying it a bit, we get:
R-S = 0.5
Alright, we've got one equation to work with. But we need another equation to find both R and S. And luckily, we have one!
Remember that the man rowed upstream for 30 minutes before realizing he lost his hat, and then it took him another 30 minutes to row back downstream to pick it up. So, the total elapsed time is 1 hour (or 60 minutes).
So, our second equation is:
(R+S) * 1 = 440
Simplifying it further, we have:
R + S = 440
Now we have a system of two equations:
R - S = 0.5 (from equation 1)
R + S = 440 (from equation 2)
Solving these equations, we find:
R = 220.25
S = 219.75
So, the rate of the stream is approximately 219.75 yards per hour, and the total elapsed time is 60 minutes.
Isn't that a "streamy" calculation? I hope you found it "hat-tastic"!
To solve this problem, let's assume:
- The speed of the man in still water is denoted by M mph.
- The speed of the stream is denoted by S mph.
Since the man is rowing upstream, the effective speed of the man against the current is (M - S) mph. When the man rows back downstream, the effective speed becomes (M + S) mph.
Given that the man rows for 30 minutes (or 0.5 hours) until he notices his lost hat at point N, we can determine the distance between M and N.
Distance = Speed × Time
Distance = (M - S) × 0.5
We are also given that the distance between M and P is 440 yards.
Distance = Speed × Time
Distance = (M + S) × Time
Since the elapsed time is the same for both cases, we can use this information to find M and S.
(M - S) × 0.5 = 440
Now let's solve this equation to find M - S:
0.5M - 0.5S = 440
M - S = 880 [1]
We can also use the second equation to find M + S:
(M + S) × Time = 440
(M + S) × Time = 440
Since Time = 0.5, we can substitute it into the equation:
(M + S) × 0.5 = 440
M + S = 880 [2]
Now we have a system of linear equations [1] and [2] with two variables (M - S and M + S).
By adding equations [1] and [2], the S terms will cancel out:
(M - S) + (M + S) = 880 + 880
2M = 1760
Simplifying the equation, we find:
M = 1760 / 2
M = 880 mph
Now we can substitute the value of M into equation [2] to find S:
880 + S = 880
S = 0 mph
Since S = 0 mph, it means that there is no stream or the stream is negligible in this problem.
Therefore, the rate of the stream is 0 mph, which means there is no stream affecting the rowing. The total elapsed time is 30 minutes (or 0.5 hours).
To find the rate of the stream and the total elapsed time, we can set up a system of equations based on the given information.
Let's say the rower's speed in still water is 'r' miles per hour, and the speed of the stream is 's' miles per hour. The rower is traveling upstream, so his effective speed will be reduced by the speed of the stream, resulting in a speed of (r - s) miles per hour.
We know that the rower takes 30 minutes (or 0.5 hours) to notice the loss of his hat and start rowing back. During this time, the hat has traveled downstream with the speed of the river for 0.5 hours, covering a distance of 0.5s miles.
When the rower starts rowing back, he will be traveling downstream, so his effective speed will be increased by the speed of the stream, resulting in a speed of (r + s) miles per hour.
Since the rower notices his hat at point N, which is 440 yards below point M, we can set up the following equation for the time it takes for the rower to row back from N to M:
440 = (r + s) * t, where 't' represents the time taken in hours.
Since the rower takes 0.5 hours to notice the loss of his hat and 0.5 hours to row back to point P, the total elapsed time is 1 hour, or 60 minutes.
To determine the values of 'r' and 's', we will solve the system of equations:
Equation 1: 0.5s = (r - s) * 0.5
Equation 2: 440 = (r + s) * 0.5
Let's solve for 'r' and 's' using these equations:
From Equation 1, we can multiply both sides by 2 to eliminate the fraction:
s = r - s
Substituting this value for 's' in Equation 2:
440 = (r + r - s) * 0.5
440 = (2r - s) * 0.5
880 = 2r - s
Now, we have a system of equations:
s = r - s
880 = 2r - s
Adding the two equations together:
s + r - s + 880 = r + r - s + 880
880 = 2r + r
880 = 3r
Dividing both sides by 3:
r = 880/3
r = 293.33
Now, let's substitute this value of 'r' into Equation 1 to solve for 's':
s = r - s
s = 293.33 - s
Adding 's' to both sides:
2s = 293.33
Dividing both sides by 2:
s = 293.33/2
s = 146.67
Therefore, the rate of the stream is 146.67 yards per hour, and the total elapsed time is 60 minutes, or 1 hour.
c = speed of current
r = rowing speed
The time taken by the hat is 440/c minutes
The the rower goes upstream a distance of 30(r-c)
then back that same distance at speed r+c
So we have
440/c = 30 + 30(r-c)/(r+c) + 440/(r+c)
Oddly enough, if the current flows at 22/3 yd/min
then the rower can have any speed greater than that.
For example, if r=44/3, then a total of 60 minutes is used.
But you can plug in other rowing speeds as well.