The Bonneville Salt Flats, located in Utah near the border with Nevada, not far

Question

The Bonneville Salt Flats, located in Utah near the border
with Nevada, not far from interstate I-80, cover an area of
over 30000 acres. A race car driver on the Flats first heads
north for 5.05 km, then makes a sharp turn and heads southwest
for 1.73 km, then makes another turn and heads east for
3.87 km. How far is she from where she started?

a) Find the magnitude of - ~A + ~B , where ~A = (22.9, 55.1),
~B= (89.5, -150.0). ?

b) Find the direction of - ~A + ~B , where ~A = (22.9, 55.1),
~B= (89.5, -150.0). Express the angle counterclockwise relative
to the positive x-axis. ?

Explanation / Answer

You've got several right triangles to solve. The 1.73 km southwest leg needs to be split into its south and west components. The 1.73 km distance is the hypotenuse of a right triangle with the south distance as one side and the west distance as the other side. Since the hypotenuse is the square root of the sum of the squares of the sides: 1.73 = sqrt(S^2 + W^2) Since the S and W components are equal, this can be written as: 1.73 = sqrt(2 * S^2) Square both sides to get: 1.73^2 = 2 * S^2 or 2.9929 = 2 * S^2 Divide both sides by 2 to get: 1.4964 = S^2 Take the square root of both sides to get: S = 1.22 (rounded to two decimal places) The southwest leg takes the driver 1.22 km south and 1.22 km west. Now you need to add the various leg of the trip. Assume north and east are positive. 5.05 km (initial north leg) -1.22 km (the south component of the southwest leg) ------------ 3.83 km (net north displacement) -1.22 km (the west component of the southwest leg) +3.87 km (the final east leg) ----------------- 2.65 km (net east displacement) The 3.83 and 2.65 km displacements are the sides of another right triangle. The hypotenuse of that triangle is the distance from the starting point. h = sqrt(3.83^2 + 2.65^2) h = 4.65 km The driver ends up 4.65 km northeast of his starting point.

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