A Texas rancher wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in the figure below, where A = 4.90 km and θC = 15°.

The answer:

the full question is as follow:

A Texas rancher wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in Figure below, where A = 4.90 km and θC = 15°. He then correctly calculates the length and orientation of the fourth side D. What is the magnitude and direction of vector D?

As shown in the figure,

A + B + C + D = 0, so to find the magnitude and direction of vector D, we should follow the following method:

D = 0 - (A + B + C),

let W = - (A + B + C), so the magnitude and direction of vector D is the same of the vector W characteristics

Magnitude

A + B + C = (4.90cos7.5 - 2.48sin16 - 3.02cos15) I

+ (-4.9sin7.5 + 2.48cos16 + 3.02sin15) J

= 1.25I + 2.53J

the magnitude of W = abs value of (A + B + C) = sqrt (1.25² + 2.53²)

= 2.82

the direction of D can be found by using Dx and Dy value

we know that tan θo = Dx / Dy = 1.25 / 2.53 = 0.49

tanθo = 0.49 it implies θo = arctan 0.49 = 26.02°

direction is 26.02°