A boat is traveling with a speed 7.2 mph over the water and at a direction of 87 degrees counterclockwise from east. The river itself is flowing east at 4.9 mph relative to its bank. What is the speed of the boat relative to the river bank to the nearest hundredth of a mph?
Perform a vector addition of the river water speed and the speed of the boar relative to the water.
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To find the speed of the boat relative to the river bank, we need to use vector addition.
Let's represent the boat's velocity as Vb and the velocity of the river as Vr. The boat is traveling at a speed of 7.2 mph in a direction 87 degrees counterclockwise from east. This can be represented as Vb = 7.2 mph at an angle of 87 degrees counterclockwise from east.
The velocity of the river is given as 4.9 mph to the east. This can be represented as Vr = 4.9 mph to the east.
To find the velocity of the boat relative to the river bank, we need to add Vb and Vr. We can use vector addition to calculate this.
First, we need to convert the boat's velocity from polar coordinates (magnitude and direction) to rectangular coordinates (x and y components).
The x component of Vb can be calculated as Vbx = Vb * cos(θ), where Vbx is the x component of Vb, Vb is the magnitude of Vb (7.2 mph), and θ is the angle of Vb (87 degrees).
The y component of Vb can be calculated as Vby = Vb * sin(θ), where Vby is the y component of Vb, Vb is the magnitude of Vb (7.2 mph), and θ is the angle of Vb (87 degrees).
Using these formulas, we can calculate Vbx and Vby:
Vbx = 7.2 mph * cos(87 degrees)
Vbx ≈ 7.2 mph * (-0.0523)
Vbx ≈ -0.3784 mph (rounded to four decimal places)
Vby = 7.2 mph * sin(87 degrees)
Vby ≈ 7.2 mph * 0.9986
Vby ≈ 7.1909 mph (rounded to four decimal places)
Now, let's denote the velocity of the boat relative to the river bank as Vbr.
Vbr = Vb + Vr
To find the x component of Vbr, we add the x components of Vb and Vr:
Vbrx = Vbx + Vr
Vbrx ≈ -0.3784 mph + 4.9 mph
Vbrx ≈ 4.5216 mph
To find the y component of Vbr, we add the y components of Vb and Vr:
Vbry = Vby + 0
Vbry = Vby
Vbry ≈ 7.1909 mph
Finally, we can calculate the magnitude (speed) of Vbr using the x and y components:
|Vbr| = sqrt(Vbrx^2 + Vbry^2)
|Vbr| = sqrt((4.5216 mph)^2 + (7.1909 mph)^2)
|Vbr| ≈ sqrt(20.4751 mph^2 + 51.7086 mph^2)
|Vbr| ≈ sqrt(72.1837 mph^2)
|Vbr| ≈ 8.49 mph (rounded to two decimal places)
Therefore, the speed of the boat relative to the river bank is approximately 8.49 mph.