Solve Applied Problems Using the Law of Cosines
-Two sailboats leave a harbor in the Bahamas at the same time. The first sails at $21 \mathrm { mph }$ in a direction $350 ^ { \circ }$. The second sails at $33 \mathrm { mph }$ in a direction $220 ^ { \circ }$. Assuming that both boats maintain speed and heading, after 4 hours, how far apart are the boats?
A) $196.8$ miles
B) $125.6$ miles
C) $142.7$ miles
D) $171.9$ miles

Question

Quizplus · Accepted Answer

The Law of Cosines states that in a triangle with sides $a$, $b$, and $c$, and opposite angles $A$, $B$, and $C$, respectively, $$c^2 = a^2 + b^2 - 2ab \cos C$$In this problem, let $a$ be the distance traveled by the first boat, $b$ be the distance traveled by the second boat, and $c$ be the distance between the two boats. We are given that $a = 21 \mathrm { mph } \times 4 \mathrm { hr } = 84 \mathrm { miles }$, $b = 33 \mathrm { mph } \times 4 \mathrm { hr } = 132 \mathrm { miles }$, and $C = 220 ^ { \circ } - 350 ^ { \circ } = -130 ^ { \circ }$. Substituting these values into the Law of Cosines, we get:$$c^2 = 84^2 + 132^2 - 2(84)(132) \cos (-130 ^ { \circ })$$$$c^2 = 6964 + 17424 + 22176 \cos (-130 ^ { \circ })$$$$c^2 = 46564 + 22176(-0.642788)$$$$c^2 = 46564 - 14272.99$$$$c^2 = 32291.01$$$$c = \sqrt{32291.01} = 179.6 \mathrm { miles }$$Therefore, the two boats are $179.6$ miles apart.

Solve Applied Problems Using the Law of Cosines -Two Sailboats 21mph21 \mathrm { mph }21mph

Solve Applied Problems Using the Law of Cosines
-Two Sailboats $21 \mathrm { mph }$