Question

solving a navigation problem
a ferry transports passengers to three different ports.
from port a, the ferry travels east 3.5 miles to port b.
the ship then turns $75^\circ$ south and travels for 6 miles to reach port c. what is the distance $d$ from port c to port a? approximate to the nearest tenth of a mile.
_______ miles

Explanation:

Step1: Identify triangle angle at B

The angle inside $\triangle ABC$ at point B is $180^\circ - 75^\circ = 105^\circ$.

Step2: Apply the Law of Cosines

The Law of Cosines states $d^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\angle ABC)$. Substitute $AB=3.5$, $BC=6$, $\angle ABC=105^\circ$:
$$d^2 = 3.5^2 + 6^2 - 2(3.5)(6)\cos(105^\circ)$$

Step3: Calculate each term

• $3.5^2 = 12.25$, $6^2 = 36$
• $\cos(105^\circ) \approx -0.2588$
• $2(3.5)(6)\cos(105^\circ) \approx 42(-0.2588) \approx -10.8696$

Step4: Compute $d^2$ and solve for d

$$d^2 = 12.25 + 36 - (-10.8696) = 48.25 + 10.8696 = 59.1196$$
$$d = \sqrt{59.1196} \approx 7.7$$

Answer:

7.7 miles