A person can row at a constant speed of 6 miles per hour. If the rower wants to row directly across a river with a current of 2 mph, at what angle should they row? How fast does the boat actually cross the river?
well, 6^2-2^2 = 32.
Did you draw a diagram?
Your angle x is such that sin(x) = 2/6
To find the angle at which the rower should row and how fast the boat actually crosses the river, we can use vector addition.
Let's denote the rower's boat velocity as Vb (which is 6 mph) and the river's current velocity as Vc (which is 2 mph). Since the current of the river affects the boat's motion, the actual velocity of the boat can be represented as the vector sum of the boat's velocity relative to the ground (Vb) and the river's current (Vc).
First, we can express the boat's velocity, Vb, as its horizontal (Vbx) and vertical (Vby) components. Since the rower wants to row directly across the river, the vertical component of the boat's velocity will be zero.
Vbx = Vb * cosθ (horizontal component of boat's velocity)
Vby = Vb * sinθ (vertical component of boat's velocity)
Where θ is the angle at which the rower should row.
To find the actual velocity of the boat (Vtotal), we can add the boat's velocity components (Vbx and Vby) and the river's current velocity (Vc).
Vtotal = Vbx + Vby + Vc
Given that Vb = 6 mph and Vc = 2 mph, substituting these values, we get:
Vtotal = (6 * cosθ) + (2 * sinθ)
To find the angle θ at which the rower should row, we can differentiate Vtotal with respect to θ and set the derivative equal to zero. Then solve for θ to find its value.
dVtotal/dθ = -6 * sinθ + 2 * cosθ = 0
Dividing both sides by 2:
-3 * sinθ + cosθ = 0
Rearranging the equation:
cosθ = 3 * sinθ
Dividing both sides by sinθ:
cotθ = 3
Taking the inverse cotangent (or arccot) of both sides:
θ = arccot(3)
Using a calculator, we find that θ is approximately 18.43 degrees.
Therefore, the rower should row at an angle of approximately 18.43 degrees to the current.
To find how fast the boat actually crosses the river, we can substitute the value of θ into the expression for Vtotal:
Vtotal = (6 * cosθ) + (2 * sinθ)
Using the value of θ ≈ 18.43 degrees:
Vtotal ≈ (6 * cos(18.43°)) + (2 * sin(18.43°))
Evaluating this expression, we find that the boat actually crosses the river at a speed of approximately 5.51 mph.