Points C and D are directly across from each other on opposite banks of a river. A boat travels across the river directly from point c to point d at a speed of 12 mph. If the current of the river has a speed of 4 mph, at what angle, and speed, must the captain head to travel directly from c to d?
he needs to head upstream at an angle θ (measured from the line CD), such that
tan θ = 4/12
so, θ = 18.4°
The speed required is √(16+144) = 12.65 mph
To determine the angle and speed at which the captain must head to travel directly from point C to point D, we can use vector addition.
Let's set up a coordinate system where the x-axis represents the direction of the river flow, and the y-axis represents the direction perpendicular to the river flow. Assume point C is at the origin (0,0).
Since the boat is directly across from point D, the distance between C and D can be considered as the horizontal component of the boat's velocity. Given that the boat's speed is 12 mph, the horizontal component of the velocity is also 12 mph.
The current of the river has a speed of 4 mph, acting in the positive x-direction. This means that the horizontal component of the boat's velocity due to the current is -4 mph.
Now, let's find the vertical component of the boat's velocity. Since the boat is directly heading from C to D, the vertical component of the velocity should be zero.
Using vector addition, we can find the resulting velocity of the boat. The horizontal and vertical components can be combined to form the resulting velocity vector. The magnitude of the resulting velocity vector represents the speed at which the captain must head, and the angle represents the direction.
Using the Pythagorean theorem, we can find the magnitude of the resulting velocity vector:
(resulting velocity)^2 = (horizontal component)^2 + (vertical component)^2
(resulting velocity)^2 = (12 mph)^2 + (0 mph)^2
(resulting velocity)^2 = 144 mph^2
Taking the square root of both sides, we find:
resulting velocity = 12 mph
So, the captain must head at a speed of 12 mph.
Since the vertical component is zero, the boat is traveling directly from C to D, which means the angle is 0 degrees or 180 degrees (depending on which side of the y-axis point D is located).
In conclusion, the captain must head directly from C to D at a speed of 12 mph and at an angle of 0 degrees or 180 degrees.