A boat is capable of 20 km/h in still water. You wish to cross the river downstream so that the angle the boat's push makes with the bank is 50 degrees. At what angle to the bank should the boat steer if the current is 8 km/h? How long will it take to cross if the river is 1.7 km/h?
A. Vr = Vb + Vc.
20[50o] = Vb - 8i,
Vb = 20[50]+8i,
Vb = 20*Cos50+20*sin50 + 8i,
Vb = 12.9 + 15.3i + 8i = 12.9 + 23.3i.
Tan A = Y/X =23.3/12.9 = 1.80620,
A = 61o.
Your last question doesn't make sense.
1.7 km wide.
Srry
B. sin50 = 1.7/d. d = 2.22 km.
50o W. of N. = 140o CCW.
Vr = Vb + Vc.
20[140] = Vb + (-8i),
Vb = 20[140] + 8i,
Vb = 20*Cos140+20*sin140 + 8i.
Vb = -15.3 + 12.9i + 8i = -15.3 + 20.9i
Tan A = Y/X = 20.9/(-15.3) = -1.36601. A = -53.8o = 53.8o N. of W. = 126.2o CCW = 36.2O W of N.
To find the angle at which the boat should steer, we need to use basic trigonometry. Let's call the angle between the boat's heading and the bank angle "θ".
First, let's analyze the forces acting on the boat when it's crossing the river downstream. The boat's speed in still water is 20 km/h, and the current is flowing at 8 km/h. So, the total speed of the boat relative to the bank is the vector sum of the boat's speed and the current's speed, which is √(20² + 8²) km/h ≈ 21.54 km/h.
Now, to find the angle θ, we can use the sine function. The sine of θ is opposite/hypotenuse, which in this case is the current's speed (8 km/h) divided by the total speed (21.54 km/h). So, sin(θ) = 8/21.54.
To find the angle θ, we can take the inverse sine (also known as arcsine) of both sides of the equation: θ = arcsin(8/21.54). Using a scientific calculator, we find that θ is approximately 21.1 degrees.
Therefore, the boat should steer at an angle of approximately 21.1 degrees to the bank.
To calculate how long it will take to cross the river, we can use the distance formula. The river's width is given as 1.7 km, and the boat's speed in still water is 20 km/h. Since the boat is moving at an angle to the bank, only a component of its speed will contribute to crossing the river. This component is the perpendicular component, which is found by multiplying the total speed (21.54 km/h) by the cosine of the steering angle.
So, the time it will take to cross the river is given by: time = distance / component of speed.
Using the given values, the time it will take to cross the river is: time = 1.7 km / (21.54 km/h * cos(21.1 degrees)).
After evaluating the expression using a calculator, we find that the time it will take to cross the river is approximately 0.107 hours, which is equivalent to about 6.42 minutes.