Question

Two sailboats started at the same location. Sailboat A traveled 5 miles west, then
turned 29° toward the north and continued for 8 miles. Sailboat B first went south
for 8 miles, then turned 51° toward the east and continued for 5 miles. Which
sailboat was farther from the starting point? Explain your reasoning.

Answer 1

The given question can be solved by applying the cosine rule. Therefore, Sailboat A is 12.61 miles away from the starting point. Thus it is the farthest from the starting point.

a. To determine the distance of sailboat A from the stating point.

Applying the cosine rule to the path traveled by sailboat A, we have:

$c^{2}$ = $a^{2}$ + $b^{2}$ - 2ab Cos θ

   = $5^{2}$ + $8^{2}$ - 2(5*8) Cos (180 - 29)

   = 25 + 64 -80 * -0.8746

$c^{2}$ = 89 + 70

$c^{2}$ = 159

c = $\sqrt{159}$

  = 12.6095

Thus, the distance of sailboat A from starting point is 12.61 miles.

b. To determine the distance of sailboat B from the stating point.

Applying the cosine rule to the path traveled by sailboat B, we have:

$c^{2}$ = $a^{2}$ + $b^{2}$ - 2ab Cos θ

   = $8^{2}$ + $5^{2}$ - 2(5*8) Cos (180 - 51)

   = 64 + 25 -80 * -0.6293

$c^{2}$ = 89 + 50.344

   = 139.344

c = $\sqrt{139.344}$

  = 11.8044

c = 11.80 miles

Thus, the distance of sailboat B from starting point is 11.80 miles.

Comparing the distances of the sailboats from the starting point, sailboat A is farther from the starting point.

A sketch is attached to this answer for better reasoning.

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