A river is 2000 ft wide and flowing at 6 mph from north to south. A woman in a canoe starts on the eastern shore and heads west at her normal paddling speed of 2 mph. In what direction (measured clockwise from north) must she aim her canoe? How long will it take her to go directly across the river? Justify and explain your reasoning.
1. tanAr = -6/-2 = 3.0,
Ar = 71.6 Deg = Reference angle.
A = 180 + Ar = 180 + 71.6 = 251.6 Deg.
Da = 180 - 71.6 = 108.4 Deg.,CCW =
Direction she should aim her canoe.
2. d = V*t,
t = d/V = (2000/5280) / 2 = 0.1894 h =
11.4 min.
The only problem here is that when drawing your reference triangle, the westward speed was 2 and the downstream speed was 6, so you could take the tangent.
When rowing upstream, the 2 mph speed is along the hypotenuse, She can't row the required 2√10 = 6.32 mph needed to end up with a westward speed of 2 mph.
the x = arctan 2/6 = 18.43
so, the direction(measured clockwise) must be 18.43+ 180= 198.4 degrees.
To find the direction the woman should aim her canoe, we need to determine the angle between the direction of the river's flow and the direction she wants to go.
First, let's draw a diagram to visualize the situation:
N
|
W ------------------- E
|
S
Here, N represents north, E represents east, S represents south, and W represents west. The woman is starting on the eastern shore and wants to head directly west. The river is flowing from north to south.
To find the angle, we need to use some trigonometry. Given the river's flow speed, we can determine the river's movement in the east-west direction. Since the river is flowing south, its east-west velocity is zero.
Using the Pythagorean theorem, we can find the speed of the river's flow in the north-south direction:
(squared of river's flow speed) = (squared of east-west velocity) + (squared of north-south velocity)
Let's call the north-south velocity of the river "v".
6^2 = 0 + v^2
36 = v^2
v = 6 mph
So, the river's north-south velocity is 6 mph.
Now, we can calculate the angle using the tangent function:
tan(angle) = (north-south velocity) / (east-west velocity)
tan(angle) = 6 / 2
tan(angle) = 3
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(3)
Using a calculator, we can find that the angle is approximately 71.57 degrees.
So, the woman must aim her canoe approximately 71.57 degrees clockwise from north to counteract the river's flow and travel directly west.
To calculate how long it will take her to go directly across the river, we can use the concept of relative velocities. By combining her paddling speed and the river's flow speed, we can determine the effective speed at which she will be moving across the river.
The effective speed can be found using the Pythagorean theorem:
(effective speed)^2 = (paddling speed)^2 + (river's flow speed)^2
(effective speed)^2 = 2^2 + 6^2
(effective speed)^2 = 4 + 36
(effective speed)^2 = 40
effective speed ≈ 6.3246 mph
Thus, the woman's effective speed across the river is approximately 6.3246 mph.
To find the time it will take her, we divide the width of the river by her effective speed:
time = (river's width) / (effective speed)
time = 2000 ft / (6.3246 mph)
time ≈ 315.97 minutes
Therefore, it will take her approximately 315.97 minutes (or about 5 hours and 16 minutes) to go directly across the river.
In summary, the woman should aim her canoe approximately 71.57 degrees clockwise from north to counteract the river's flow and travel directly west. It will take her approximately 5 hours and 16 minutes to go directly across the river.