Answer:
To find the resultant velocity and direction of the boat, we need to use vector addition.
Let's consider the velocity of the boat as a vector in the east direction, with a magnitude of mph. We can represent this vector as follows:
v1 = mph, due east
Now let's consider the velocity of the wind as a vector in the northwest direction, with a magnitude of 10 mph. We can represent this vector as follows:
v2 = 10 mph, 45 degrees north of west
To find the resultant velocity, we can add the two vectors together using vector addition. We can break each vector into its x and y components as follows:
v1x = mph, v1y = 0
v2x = -7.07 mph, v2y = 7.07 mph
The negative sign in front of v2x indicates that the wind is blowing in the opposite direction to the boat's motion.
Now we can add the x and y components separately to get the resultant vector:
vx = v1x + v2x = 6.93 mph, east of north
vy = v1y + v2y = 7.07 mph, north
The magnitude of the resultant velocity is:
|v| = sqrt(vx^2 + vy^2) = sqrt((6.93 mph)^2 + (7.07 mph)^2) = 9.99 mph
The direction of the resultant velocity can be found by taking the inverse tangent of the ratio of the y-component to the x-component:
θ = tan^(-1)(vy/vx) = 45.03 degrees north of east
Therefore, the resultant velocity of the boat is 9.99 mph, 45.03 degrees north of east.
Step-by-step explanation: