Solve the River Current Mystery: Boat Speed & Water Current Calculation

[PoetryInMotion]

Okay, this may sound silly... but it's driving me crazy! Some time ago, my friend showed me an extra credit problem for her class and ever since I haven't been able to figure it out. Now classes are over and we've all moved home, but it's still bugging me to no end. So, if you can help... please do!


A boat starts traveling down a river on a boat at point A and at point B a hat falls out of the boat (and floats), but the driver doesn't realize that the hat fell out until he reaches point C. The man turns around the boat and meets up with the hat at point D.

The distance between point A and B is 1 mile.
The time it takes to travel from point B to C is 20 minutes.
It takes the same amount of time for the boat to travel to point C to D as it takes the hat to travel from point B to point D.
The boat is traveling at a constant speed.

What is the current of the water?

And here's a little diagram of the picture her teacher drew them.

C (current is headed downward)



B



A D

Help!

 

[HallsofIvy]

"It takes the same amount of time for the boat to travel to point C to D as it takes the hat to travel from point B to point D."

Surely that's not true. The hat didn't sit at point B waiting for 20 minutes until the boat reached point C and then start floating downstream as soon as the boat turned back upstream! What is true is that the time taken for the boat to go down stream from B to C and then back from C to D is the same as the time taken for the hat to float from B to D. The boat and hat were together at point B and then again at point D. The time elapsed is the same for both.

Call the speed of the boat, relative to the water, u and the current speed s. Since the boat is initially traveling down stream, its speed, relative to the bank, is u+ v. When it turns and goes back up stream, its speed, relative to the bank, is u- v. Of course, the speed of the hat, relative to the bank, once it has fallen out of the boat is the same as the speed of the water: v.

When the hat falls out of the boat, the boat continues downstream for an unknown time t₁ at speed u+v to point C: it travels a distance (u+v)t₁. It then turns around and travels upstream for another unknown time t₂ at speed u-v to point D: it travels a distance (u-v)t₂. The actual distance from point B to point D is (u+v)t₁- (u-v)t₂= u(t₁- t₂)+ v(t₁+ t₂).

During that time, the hat is floating downstream at speed v for time t₁+ t₂. The distance it floats, which is also the distance from B to D is v(t₁+ t₂.

Setting those two different calculations for the distance from B to D equal:
u(t₁- t₂)+ v(t₁+ t₂)= v(t₁+ t₂.
Notice that the "v(t₁+ t₂)" terms on each side cancel leaving us with u(t₁- t₂)= 0 or t₁= t₂. The time going downstream must be equal to the time going upstream!

We can now note that t₁ is given as 20 minutes so t₂ is also 20 minutes.

Unfortunately, we still have no way to find either u or v separately. The information that "The distance between point A and B is 1 mile" doesn't help at all since that has nothing to do with the hat.

If we were told "the distance between point B and point D" is 1 mile, then, since we know it took the boat 20 minutes to go from point B to point C, and that is the same as the time the to go back up from C to D, we would know that the hat was floating for 40 minutes. We could then calculate that v= 1 mile/40 minutes= 1.5 miles per hour.

 

[M98Ranger]

First of all, don't worry, this is not a silly question at all! It's actually a great problem to practice your problem-solving skills.

To solve this mystery, we need to use the formula: distance = speed x time.

We know that the boat travels 1 mile from point A to point B and it takes 20 minutes to travel from point B to C. This means that the boat's speed is 1 mile / 20 minutes = 0.05 miles per minute.

Now, let's look at the hat. We know that it takes the same amount of time for the hat to travel from point B to D as it takes for the boat to travel from point C to D. This means that the hat's speed is the same as the boat's speed, which is 0.05 miles per minute.

We also know that the distance from point B to D is the same as the distance from point C to D, which is the length of the boat. Let's call this distance "x".

Using the formula, we can write the following equations:

Distance boat travels = 0.05 miles/minute x 20 minutes = 1 mile

Distance hat travels = 0.05 miles/minute x x minutes = x miles

Since the boat and the hat meet at point D, the total distance traveled by both of them is 1 mile. This means that we can set the two equations equal to each other and solve for x:

1 mile = x miles

x = 1 mile

So, the distance from point B to D (which is also the distance from point C to D) is 1 mile.

Now, we can use this information to find the current of the water. Remember, the current is the speed of the water.

We know that the boat's speed is 0.05 miles per minute. But, since the boat is traveling against the current, we need to subtract the current's speed from the boat's speed.

So, we can write the following equation:

0.05 miles per minute - current's speed = 0 miles per minute

We know that the distance from point C to D is 1 mile and it takes the boat and the hat the same amount of time to travel this distance. So, we can write another equation:

0.05 miles per minute - current's speed = 1 mile / (

 

[M98Ranger]

First of all, don't worry about feeling silly for not being able to solve this problem. It's completely normal to get stuck on a problem and need some help figuring it out. Let's break this down and see if we can solve the mystery of the river current.

We know that the boat is traveling at a constant speed, which means that the distance it travels is directly proportional to the time it takes. In other words, if the boat travels twice as far, it will take twice as long. This is a key piece of information that we will use to solve the problem.

Next, we need to understand the relationship between the boat and the hat. The problem tells us that it takes the same amount of time for the boat to travel from point C to D as it does for the hat to travel from point B to D. This means that the boat and the hat are traveling at the same speed, but in opposite directions. This also tells us that the distance from point C to D is the same as the distance from point B to D.

Now, let's look at the time it takes for the boat to travel from point B to C. We know that it takes 20 minutes, but we don't know the distance. However, we do know that the distance from point A to B is 1 mile. Since the boat is traveling at a constant speed, we can use the time and distance to calculate the speed of the boat using the formula speed = distance/time. In this case, the speed of the boat is 1/20 miles per minute.

Since we know the speed of the boat, we can use this information to solve for the current of the water. Remember, the boat and the hat are traveling at the same speed, but in opposite directions. This means that the current must be the difference between the speed of the boat and the speed of the hat. In other words, the current is 1/20 - 1/20 = 0 miles per minute. This means that the current is not affecting the movement of the boat or the hat at all.

In conclusion, the mystery of the river current is solved. The current of the water is 0 miles per minute, as it is not affecting the movement of the boat or the hat. I hope this helps put your mind at ease and you can finally move on from this puzzling problem. Keep up the good work and don't be afraid to ask for help when