The distance between airports A and B is approximately 401.18 miles
How to determine the distance between airports A and B?
To find the distance between airports A and B, we can use the law of cosines since we have a triangle formed by the initial path (320 miles), the additional path (112 miles), and the angle between them when the airplane adjusts its course.
Let's denote the distance between airports A and B as 'd'. Using the law of cosines:
![\[ d^2 = 320^2 + 112^2 - 2 * 320 * 112 * \cos(180\textdegree - (350\textdegree - 300\textdegree) \]](https://img.qammunity.org/2024/formulas/physics/high-school/1fu7k9x2mw15wpuni4uyp18808oq3g078o.png)
First, let's find the angle between the two legs of the triangle:
![\[ 180\textdegree - (350\textdegree - 300\textdegree) = 180\textdegree - 50\textdegree = 130\textdegree \]](https://img.qammunity.org/2024/formulas/physics/high-school/642jafnvhtfwyx5lcdp3x55zqqmmxw23ux.png)
Now, substitute this into the law of cosines equation:
![\[ d^2 = 320^2 + 112^2 - 2 * 320 * 112 * \cos(130\textdegree) \]](https://img.qammunity.org/2024/formulas/physics/high-school/j7z8qu6zoj3wq21o8v3nm3ik1yzx16lsgf.png)
Calculate the cosine of 130\textdegree:
![\[ \cos(130\textdegree) \approx -0.6428 \]](https://img.qammunity.org/2024/formulas/physics/high-school/eu9cgcqcil5pfzad95cizlshmqrj865es2.png)
Now, substitute this into the equation:
![\[ d^2 = 320^2 + 112^2 - 2 * 320 * 112 * (-0.6428) \]](https://img.qammunity.org/2024/formulas/physics/high-school/dfed59ymbz6o20t1hvq1xkhk4unxuvakxk.png)
![\[ d^2 = 102400 + 12544 + 46006.848 \]](https://img.qammunity.org/2024/formulas/physics/high-school/misgb9h9xmg23x16wywtd94nudviqcfwb2.png)
![\[ d^2 = 160950.848 \]](https://img.qammunity.org/2024/formulas/physics/high-school/6hzc2th2jw4miy8fk3l81mw6enxh7z8862.png)
Finally, take the square root to find 'd':
![\[ d \approx √(160950.848) \]](https://img.qammunity.org/2024/formulas/physics/high-school/iil5ph8wr1ccybbimmaewckgf52wo328xw.png)
![\[ d \approx 401.18 \]](https://img.qammunity.org/2024/formulas/physics/high-school/wlh2brt3ly1fxhd9lo4citlpq0mlm8hd3m.png)
Therefore, the distance between airports A and B is approximately 401.18 miles.
See the image below for complete question.