Boat crossing river perpendicular at 30° angle heading upstream

[egg man]

a river flows at a speed Vr=5.84 km/hr with respect to shore. a boat needs to go perpendicular to the shoreline to reach a pier across the river. the boat heads upstream at an angle of 30 degrees. if the time taken for the boat to cross is 15.3 min, what is the width of the river?

okay, i know the actual velocity of the boat over the velocity of the river(5.84) equals 1/sin30. so the velocity of the boat equals 11.68 km/hr. i divide by 60 and i get .195km/min. i multiply by 15.3 and get 2.98km. but that's not right. Pleeaaaase help

 

[HallsofIvy]

What I would do is break the vector velocities into "components". If the boat is aimed at 30 degrees (to the perpendicular) with speed Vr, then the x and y components of the velocityh vector are Vr cos(30) and Vr sin(30) respectively.
At that speed, the distance he goes (across the river- ignore the up or downstream motion) is Vr cos(30)*t= (5.84)(cos(30))(15.3).

Notice that the speed of the river doesn't come into this! That would, of course, affect what angle we have to aim upstream but we are, apparently, told that this was 30 degrees.

 

[M98Ranger]

First of all, great job on calculating the velocity of the boat! That is an important step in solving this problem. However, it seems like you may have made a small mistake in your final calculation. Let's break down the steps to find the width of the river:

1. Convert the boat's velocity from km/hr to km/min: Since we are given the time in minutes, it would be easier to have the boat's velocity in km/min. So, we divide 11.68 km/hr by 60 to get 0.195 km/min.

2. Use the time and boat's velocity to find the distance traveled by the boat: We know that the boat traveled for 15.3 minutes at a velocity of 0.195 km/min. So, we can multiply these two numbers to get the distance traveled by the boat, which is 2.98 km.

3. Use trigonometry to find the width of the river: Now, we can use the distance traveled by the boat, the velocity of the river, and the angle of 30 degrees to set up a right triangle. The distance traveled by the boat is the adjacent side, the velocity of the river is the opposite side, and the width of the river is the hypotenuse. So, we can use the formula tan(theta) = opposite/adjacent to find the width of the river.

tan(30 degrees) = 5.84 km/hr / 0.195 km/min

tan(30 degrees) = 29.9

width of the river = opposite side = 29.9 * 2.98 km = 89.1 km

So, the width of the river is approximately 89.1 km. I hope this helps clarify where you went wrong in your calculation. Keep up the good work!