A woman walks due west on the deck of a ship at 4 mi/h. The ship is moving north at a speed of 15 mi/h. Find the speed and direction of the woman relative to the surface of the water. (Round your answers to one decimal place.)
To find the speed and direction of the woman relative to the surface of the water, we can use vector addition.
Let's denote the velocity of the woman walking due west as vector A and the velocity of the ship moving north as vector B.
Vector A has a magnitude of 4 mi/h and is directed due west (which we can represent as -4i, since "west" is in the negative x direction).
Vector B has a magnitude of 15 mi/h and is directed due north (which we can represent as 15j, since "north" is in the positive y direction).
To find the resultant velocity (vector R) of the woman's motion relative to the water's surface, we add vectors A and B:
R = A + B
R = (-4i) + (15j)
To calculate the magnitude of R (the speed of the woman relative to the water's surface), we use the Pythagorean theorem:
|R| = sqrt((-4)^2 + (15)^2)
|R| = sqrt(16 + 225)
|R| = sqrt(241)
|R| ≈ 15.52 mi/h (rounded to one decimal place)
To find the direction of R, we can use trigonometry. The direction angle (θ) can be calculated using the inverse tangent function:
θ = arctan(15/4)
θ ≈ 75.96 degrees (rounded to one decimal place)
Thus, the speed of the woman relative to the water's surface is approximately 15.52 mi/h, and her direction is approximately 75.96 degrees north of west.