A boy can row a boat at a constant rate of 5 mi/hr in still water. He rows upstream for 19 minutes and then rows downstream, returning to his starting point in another 13 minutes. What is the rate of the current? What is the total distance traveled?
let the speed of the current be x mph
so rate upstream = 5-x mph
and rate downstream = 5+x mph
then
19/60 (5-x) = 13/60 (5+x)
19(5-x) = 13(5+x)
95 - 19x = 65 + 13x
30 = 32x
x = 30/32 mph or .9375 mph
total distance = 13/60(5.9375) = 1.296 miles each way.
check:
rate against current = 4.0625
rate with the current = 5.9375
time against current = 1.296/4.0625 = .316 hrs = 19.4 minutes
time with current = 1.296/5.9375 = .2183 = 13.1 minutes
(Acceptable round-off error)
Well, well, well, looks like we have a little boat adventure on our hands! Let me put on my sailor hat and help you out!
Alright, so let's break it down. The boy rows upstream for 19 minutes, which means he's rowing against the current. And then he rows downstream for 13 minutes, which means he's rowing with the current.
Now, we know that the boy's rowing rate in still water is 5 mi/hr. Let's call the rate of the current "c".
When he's rowing upstream, his effective speed is reduced by the current, so his speed is 5 - c mi/hr. And when he's rowing downstream, the current helps him, so his speed is 5 + c mi/hr.
Since distance = speed x time, we can use this information to set up an equation.
In the 19 minutes rowing upstream, he goes a certain distance. Let's call that distance "d1". So, d1 = (5 - c) * (19/60) mi.
In the 13 minutes rowing downstream, he goes the same distance back to his starting point. Let's call that distance "d2". So, d2 = (5 + c) * (13/60) mi.
Since he ends up back where he started, d1 + d2 = 0.
Now we have two equations. Let's put on our math cap and solve them!
d1 = (5 - c) * (19/60)
d2 = (5 + c) * (13/60)
d1 + d2 = 0
Now, let's add up d1 and d2:
(5 - c) * (19/60) + (5 + c) * (13/60) = 0
Got it? Good! Now it's just a matter of solving for "c" to find the rate of the current.
And as for the total distance traveled, it would be the sum of d1 and d2.
Now, I don't want to spoil all the fun by giving away the answer too soon. So, grab a pen and scratch that sailor beard, and go ahead and solve those equations! I believe in you!
To find the rate of the current and the total distance traveled, we need to set up some equations based on the information given.
Let's assume "x" represents the rate of the current.
When the boy rows upstream, his effective speed will be slowed down by the current. So the speed of the boat relative to the water (still water speed - current speed) will be (5 - x) mi/hr.
When the boy rows downstream, the current will assist his speed. So the speed of the boat relative to the water (still water speed + current speed) will be (5 + x) mi/hr.
Now, let's calculate the distances for each leg of the trip.
Distance = Speed * Time (note that the time needs to be converted to hours)
For the upstream leg:
Distance_upstream = (5 - x) * (19/60) miles
For the downstream leg:
Distance_downstream = (5 + x) * (13/60) miles
Since the total distance traveled is the sum of the distances upstream and downstream, we can set up the equation:
Total distance = Distance_upstream + Distance_downstream
Total distance = (5 - x) * (19/60) + (5 + x) * (13/60)
Now, let's solve for x by using the information that the total distance is zero (since the boy returns to his starting point).
0 = (5 - x) * (19/60) + (5 + x) * (13/60)
0 = (95 - 19x + 65 + 13x)/60
0 = (160 - 6x)/60
0 = 160 - 6x
6x = 160
x = 160/6
x ≈ 26.67/6
x ≈ 4.44 mi/hr
Therefore, the rate of the current is approximately 4.44 mi/hr.
To find the total distance traveled, we can substitute the value of x back into the equation for the total distance:
Total distance = (5 - 4.44) * (19/60) + (5 + 4.44) * (13/60)
Total distance = 0.56 * (19/60) + 9.44 * (13/60)
Total distance = 0.56 * 19/60 + 9.44 * 13/60
Total distance ≈ 0.37 + 2.04
Total distance ≈ 2.41 miles
Therefore, the total distance traveled is approximately 2.41 miles.
To find the rate of the current and the total distance traveled, we can use the concept of relative speed.
Let's assume the rate of the current is 'c' mi/hr.
When rowing upstream, the effective speed of the boat is reduced by the speed of the current, so its speed is (5 - c) mi/hr.
When rowing downstream, the effective speed of the boat is increased by the speed of the current, so its speed is (5 + c) mi/hr.
We are given that the boy rows upstream for 19 minutes and downstream for 13 minutes, which we can convert to hours:
19 minutes = 19/60 hours = 0.3167 hours
13 minutes = 13/60 hours ≈ 0.2167 hours
Now, let's calculate the distances traveled:
Distance upstream = Speed x Time = (5 - c) x 0.3167
Distance downstream = Speed x Time = (5 + c) x 0.2167
Since the total distance traveled is the sum of the distances upstream and downstream, we have:
Total distance = Distance upstream + Distance downstream
= (5 - c) x 0.3167 + (5 + c) x 0.2167
Since the total distance traveled is equal to zero (the boy returns to his starting point), we have:
(5 - c) x 0.3167 + (5 + c) x 0.2167 = 0
Now, we can solve this equation to find the value of 'c' (the rate of the current):
(5 - c) x 0.3167 + (5 + c) x 0.2167 = 0
0.3167(5 - c) + 0.2167(5 + c) = 0
1.5835 - 0.3167c + 1.0835 + 0.2167c = 0
2.667 = 0.1c
c = 2.667 / 0.1
c ≈ 26.67 / 1
c ≈ 2.67 mi/hr
Therefore, the rate of the current is approximately 2.67 mi/hr.
To find the total distance traveled, we can substitute the value of 'c' into the distance equation:
Total distance = (5 - 2.67) x 0.3167 + (5 + 2.67) x 0.2167
= 2.33 x 0.3167 + 7.67 x 0.2167
≈ 0.7372 + 1.6632
≈ 2.4004 mi
Therefore, the total distance traveled is approximately 2.4004 miles.