An east west river is 2km wide and flows at 4 km/h E. A motorboat has a speed of 10km/h in still water. The motorboat heads due north from the south bank of the river. There is a restaurant directly across the river on the north bank from where the boat starts. How far from the restaurant will the boat be when it reaches the north shore AND how long did it take to get there?
someone help fast with the solution plsss
time for crossing ... 2 km / 10 km/hr = .2 hr
distance from restaurant ... 4 km/hr * .2 hr = .8 km
Well, well, well! We have ourselves a river-crossing adventure, don't we? Don't worry, my friend, Clown Bot is here to bring in some fun math equations!
First, let's break it down. We have an east-west river that is 2km wide, flowing at 4 km/h towards the east. Our mighty motorboat, on the other hand, zooms along at 10 km/h in still waters.
Since the boat is heading due north from the south bank, we need to analyze the boat's speed with respect to the river's flow. If the river flows east at 4 km/h and the boat's speed in still water is 10 km/h, they combine to give the boat an effective speed of 14 km/h towards the east (10 + 4 = 14). Think of it like the river pushing the boat along, giving it an extra push!
Now, it's time to calculate the time it takes for the boat to reach the north shore. We can do this by dividing the distance the boat needs to travel by its effective speed.
The boat needs to cross the 2km wide river, so let's divide that by the boat's effective speed of 14 km/h. 2 km / 14 km/h = 0.14285714285714285 hours.
Now we have the time it takes for the boat to reach the north shore, but let's convert it into a more understandable format. Since there are 60 minutes in an hour, our boat ride will take approximately 0.14285714285714285 x 60 = 8.57 minutes.
So, the boat will be approximately 2 km away from the restaurant on the north shore when it reaches the north shore. It will take about 8.57 minutes or 8 minutes and 34 seconds to get there.
Now, go forth, my river-crossing friend, and enjoy your boat ride to the restaurant! Bon voyage!
To find the distance from the restaurant when the boat reaches the north shore, we need to calculate the boat's drift caused by the river's current and the boat's relative speed to the north.
Let's break down the problem step by step:
Step 1: Calculate the boat's drift caused by the river's current.
- The river flows eastward at a speed of 4 km/h.
- Since the boat is heading north, only the eastward component of the river's flow will affect the boat's position.
- The boat's drift caused by the river's current can be calculated using the formula: drift = speed of the river * time.
- In this case, since the boat is heading north across a 2 km wide river, the drift will be 2 km (width of the river).
Step 2: Calculate the boat's effective speed.
- The boat has a speed of 10 km/h in still water.
- Since the boat is heading north and there is a drift of 2 km to the east caused by the river, the boat's effective speed can be calculated using the Pythagorean theorem.
- The effective speed of the boat can be calculated as the square root of (boat's speed in still water)^2 + (boat's drift caused by the river)^2.
- In this case, the effective speed of the boat can be calculated as the square root of (10 km/h)^2 + (2 km)^2.
Step 3: Calculate the time taken by the boat to reach the north shore.
- We know the distance to be covered by the boat is the width of the river, which is 2 km.
- We have the effective speed of the boat, which is obtained in Step 2.
- Time can be calculated using the formula: time = distance / speed.
- In this case, the time taken by the boat to reach the north shore can be calculated as 2 km / effective speed.
Step 4: Calculate the distance from the restaurant when the boat reaches the north shore.
- The boat is heading north from the south bank, so the distance from the restaurant will be the same as the distance from the boat's starting point on the south bank.
- Since the boat is heading directly north, the distance from the restaurant will be the same as the width of the river, which is 2 km.
Let's calculate the answers:
Step 1: Drift caused by the river's current:
drift = 4 km/h * time (unknown)
drift = 2 km
Step 2: Effective speed of the boat:
effective speed = sqrt((10 km/h)^2 + (2 km)^2)
Step 3: Time taken by the boat to reach the north shore:
time = 2 km / effective speed (unknown)
Step 4: Distance from the restaurant when the boat reaches the north shore:
distance = width of the river = 2 km
To solve Step 2, calculate the effective speed of the boat. Then substitute the effective speed into Step 3 to find the time taken by the boat. Finally, in Step 4, substitute the width of the river into the distance formula.
To solve this problem, we can use the principles of relative velocity, which means considering the velocities of the boat and the river relative to each other.
1. Determine the velocity of the boat relative to the water:
Since the boat has a speed of 10 km/h in still water and is heading due north, the velocity of the boat relative to the water is 10 km/h due north.
2. Determine the velocity of the river:
The river flows at 4 km/h towards the east. Since we are considering it relative to the boat's motion, the velocity of the river relative to the boat is in the opposite direction, which is 4 km/h towards the west.
3. Calculate the net velocity of the boat:
To find the net velocity of the boat, we need to add the velocity of the boat relative to the water and the velocity of the river relative to the boat. In this case, the velocities are perpendicular to each other, so we can use the Pythagorean theorem to find the net velocity.
Net velocity = √((velocity of boat relative to water)² + (velocity of river relative to boat)²)
Net velocity = √((10 km/h)² + (4 km/h)²)
Net velocity = √(100 km²/h² + 16 km²/h²)
Net velocity = √116 km²/h²
Net velocity ≈ 10.77 km/h
4. Calculate the time taken to cross the river:
The width of the river is given as 2 km, and the boat's net velocity is 10.77 km/h. We can use the formula: time = distance / speed to find the time taken to cross.
Time = 2 km / 10.77 km/h
Time ≈ 0.1854 hours
5. Calculate the distance from the restaurant:
Since the boat is heading due north and the net velocity is perpendicular to the river, the distance traveled north is the same as the distance from the restaurant.
Distance = velocity of boat relative to water * time
Distance = 10 km/h * 0.1854 hours
Distance ≈ 1.854 km
Therefore, the boat will be approximately 1.854 km from the restaurant when it reaches the north shore, and it will take approximately 0.1854 hours (or 11.12 minutes) to get there.