a man is driving his motor boat at 12 mph travelling N 30 E. The river currents are flowing due East at 8 mph. (A) express the true velocity of the motorboat as a vector. (B) what is the true speed of the motorboat? (C) What is the true direction of the motorboat?
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To find the true velocity of the motorboat, we need to consider the combined effect of the boat's velocity and the river's current. Let's break down the problem into two components: one in the north-south direction and the other in the east-west direction.
(A) To express the true velocity of the motorboat as a vector, we add the boat's velocity vector and the river's current vector.
Given:
Boat's velocity = 12 mph at N 30 E
River's current = 8 mph due east
First, convert the boat's velocity to its component vectors:
The speed in the east-west direction = 12 mph * cos(30°) = 12 mph * √3/2 = 6√3 mph
The speed in the north-south direction = 12 mph * sin(30°) = 12 mph * 1/2 = 6 mph
Now we have the components of the boat's velocity vector:
Boat's velocity vector = (6√3 mph, 6 mph)
Adding the river's current vector:
River's current vector = (8 mph, 0 mph)
To find the true velocity of the motorboat, we add the corresponding components:
True velocity vector = (boat's velocity vector) + (river's current vector)
= (6√3 mph + 8 mph, 6 mph + 0 mph)
= (6√3 + 8) mph, 6 mph)
Therefore, the true velocity of the motorboat is approximately (14.4 mph, 6 mph).
(B) To find the true speed of the motorboat, we calculate the magnitude of the true velocity vector:
True speed of the motorboat = √[(14.4 mph)^2 + (6 mph)^2]
= √(207.36 mph^2 + 36 mph^2)
= √(243.36 mph^2)
= 15.6 mph (rounded to one decimal place)
Therefore, the true speed of the motorboat is approximately 15.6 mph.
(C) To find the true direction of the motorboat, we use the inverse tangent function to calculate the angle formed by the true velocity vector with respect to the east-west direction:
True direction = arctan[(boat's velocity in north-south direction) / (boat's velocity in east-west direction)]
= arctan(6 mph / 6√3 mph)
≈ arctan(1 / √3)
≈ 30° (rounded to the nearest degree)
Therefore, the true direction of the motorboat is approximately N 30° E.