A ferry boat departed from port and traveled at a rate of 10 miles per hour at an angle 30° north of East across a large river. If the current is flowing at a rate of 4 miles per hour straight south, what is the speed and direction of the boat in the water? Round any irrational answers to one decimal place.
(5√3,5)+(0,-4) = (5√3,1)
To find the speed and direction of the boat in the water, we will need to break down the velocity vectors into their x and y components.
First, let's break down the velocity of the boat. The boat is traveling at a rate of 10 miles per hour at an angle 30° north of East. We can break down this velocity vector into its x and y components using trigonometry.
The x-component of the boat's velocity can be found by multiplying the magnitude of the velocity (10 mph) by the cosine of the angle. So, the x-component is:
10 mph * cos(30°)
The y-component of the boat's velocity is found by multiplying the magnitude of the velocity (10 mph) by the sine of the angle. So, the y-component is:
10 mph * sin(30°)
The current is flowing at a rate of 4 miles per hour straight south. Since the current is flowing in the opposite direction of the y-component of the boat's velocity, we subtract 4 mph from the y-component.
Now, let's calculate the x and y components of the boat's velocity in the water:
x-component of boat's velocity = 10 mph * cos(30°) = 10 mph * 0.866 = 8.66 mph (rounded to two decimal places)
y-component of boat's velocity = 10 mph * sin(30°) - 4 mph = 10 mph * 0.5 - 4 mph = 5 mph - 4 mph = 1 mph
To find the speed of the boat in the water, we can use the Pythagorean theorem. The speed is the magnitude of the resultant velocity vector.
speed = sqrt((x-component)^2 + (y-component)^2)
speed = sqrt((8.66 mph)^2 + (1 mph)^2)
speed = sqrt(75 + 1)
speed = sqrt(76)
speed ≈ 8.7 mph (rounded to one decimal place)
To find the direction of the boat in the water, we can use the inverse tangent function to calculate the angle.
direction = arctan(y-component / x-component)
direction = arctan(1 mph / 8.66 mph)
direction ≈ 0.1151 radians or 6.59° (rounded to two decimal places)
Therefore, the speed of the boat in the water is approximately 8.7 mph, and the direction is approximately 6.59° with respect to the positive x-axis.