Answer:
To solve this problem, we can use vector addition. We can represent the velocity of the boat as a vector pointing east with magnitude 7 mph, and the velocity of the wind as a vector pointing northwest with magnitude 10 mph.
To add these two vectors, we can break the wind velocity vector into its x and y components, using trigonometry. Since the vector is pointing northwest, we know that it makes a 45 degree angle with both the x and y axes. Therefore, the x component is:
10 mph * cos(45) = 7.07 mph
and the y component is:
10 mph * sin(45) = 7.07 mph
Now we can add the x component of the wind velocity vector to the velocity of the boat, since they are both pointing in the same direction (east):
7 mph + 7.07 mph = 14.07 mph
The y component of the wind velocity vector is pointing north, while the velocity of the boat is pointing east, so we cannot simply add them. Instead, we can use the Pythagorean theorem to find the magnitude of the resultant vector:
sqrt(14.07^2 + 7.07^2) = 15.78 mph
The direction of the resultant vector can be found using trigonometry. The angle between the resultant vector and the east direction is:
tan^-1(7.07/14.07) = 26.56 degrees
Therefore, the boat is sailing at a speed of 15.78 mph in a direction that is 26.56 degrees north of east.
Step-by-step explanation: