Chapter 8, Problem 34RE

To determine

To find:

a. How far is the sailboat from the island at this time?

Answer to Problem 34RE

Solution:

a. $131.8$ mi

Explanation of Solution

Given:

A sailboat leaves St. Thomas bound for an island in the British West Indies, 200 miles away. Maintaining a constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours, that the sailboat is off course by $15°$.

Formula Used:

$c^2=a^2+b^2-2ab \text{cos}C$

Calculation:

constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours. Therefore, distance covered $=4*18=72$ miles.

a. Using Law of cosines,

$x^2={200}^2+{72}^2-2*72*200*\text{ cos}15$

$x^2=17365$

$x=131.8$ mi

To determine

To find:

b. Through what angle should the sailboat turn to correct its course?

Answer to Problem 34RE

Solution:

b. ${23.1}^{\circ }$

Explanation of Solution

Given:

A sailboat leaves St. Thomas bound for an island in the British West Indies, 200 miles away. Maintaining a constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours, that the sailboat is off course by $15°$.

Formula Used:

$c^2=a^2+b^2-2ab \text{cos}C$

Calculation:

constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours. Therefore, distance covered $=4*18=72$ miles.

b. using law of sines,

$\frac{200}{\text{sin}A}=\frac{131.8}{\text{sin}15}$

$A={\text{ sin}}^{-1}(200*\sin 15/131.8)=23.1°$

To determine

To find:

c. How much time has been added to the trip because of this? (Assume that the speed remains at 18 miles per hour.)

Answer to Problem 34RE

Solution:

c. $0.21$ hours

Explanation of Solution

Given:

A sailboat leaves St. Thomas bound for an island in the British West Indies, 200 miles away. Maintaining a constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours, that the sailboat is off course by $15°$.

Formula Used:

$c^2=a^2+b^2-2ab \text{cos}C$

Calculation:

constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours. Therefore, distance covered $=4*18=72$ miles.

c. Instead of covering 200mi, the trip will be $72+131.8=203.8$ mi

$203.8-200=3.8$ mi

$3.8$ mi will be covered in $3.8/18=0.21$ hours.