Chapter 8.3, Problem 44AYU

Navigation An airplane flies due north from Ft. Myers to Sarasota, a distance of 150 miles, and then turns through an angle of $50°$ and flies to Orlando, a distance of 100 miles. See the figure.

a. How far is it directly from Ft. Myers to Orlando?

b. What bearing should the pilot use to fly directly fron Ft. Myers to Orlando?

To determine

To find:

a. How far is it directly from Ft. Myers to Orlando?

b. What bearing should the pilot use to fly directly from Ft. Myers to Orlando?

Answer to Problem 44AYU

Solution:

a. The direct distance from Ft. Myers to Orlando is approximately $227.56$ miles.

b.The pilot should fly at a bearing of $\text{N}$ $19.7° E.$

Explanation of Solution

Given:

An airplane flies due north from Ft. Myers to Sarasota, a distance of 150 miles, and then turns through an angle of $50°$ and flies to Orlando, a distance of 100 miles. See the figure.

Formula used:

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

$\frac{\text{sin}A}{a} = \frac{\text{sin}B}{b} = \frac{\text{sin}C}{c}$

Calculation:

a. To find how far is it directly from Ft. Myers to Orlando, we need to find the 3^rd side using cosine formula.

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

$a = 150, b = 100, C = 180 - 50 = 130$

$c^2 = {150}^2 + {100}^2 - 2 * 150 * 100\text{cos}130$

$c \approx 227.56 \text{mi}$

The direct distance from Ft. Myers to Orlando is approximately $227.56$ miles.

b. To find What bearing should the pilot use to fly directly from Ft. Myers to Orlando, we need to find the angle A, that is angle at Ft. Myers.

$\frac{\text{sin}A}{a} = \frac{\text{sin}B}{b} = \frac{\text{sin}C}{c}$

$\frac{\text{sin}A}{100} = \frac{\text{sin}C}{c}$

$Sin A = 100 * \frac{\text{sin}130}{227.56}$

$A = {\text{sin}}^{ -1}100 * \text{sin}130/227.56 = 19.7°$

Since the angle is $19.7°$, the pilot should fly at a bearing of $\text{N}19.7°E$.