Question

1. the navigator of a ship at sea sees a lighthouse due north of the ship. the ship then sails 3.2 km due west. the angle between the ships path and the line of sight to the lighthouse is 39.1°. how far is the ship from the lighthouse to the nearest tenth of a kilometre? a. 5.1 km b. 2.5 km c. 3.9 km d. 4.1 km 9. a ladder is 7 m long. it leans against a house. the base of the ladder is 2 m from the house. what is the angle of inclination of the ladder to the nearest tenth of a degree?

Explanation:

Step 1: Identify the trigonometric relationship for question 9

We have a right - triangle situation where the hypotenuse of the right - triangle is the length of the ladder ($c = 7$ m) and the adjacent side to the angle of inclination is the distance from the base of the ladder to the house ($a=2$ m). We use the cosine function $\cos\theta=\frac{a}{c}$.

Step 2: Calculate the cosine of the angle

$\cos\theta=\frac{2}{7}\approx0.2857$.

Step 3: Find the angle

$\theta=\cos^{- 1}(0.2857)$. Using a calculator, $\theta\approx73.4^{\circ}$.

For question 8:

Step 1: Use the law of cosines

Let the initial distance from the lighthouse to the ship be $d_1$, the distance the ship sails be $a = 3.2$ km, and the final distance from the lighthouse to the ship be $d_2$. The angle between the initial and new path of the ship is $\beta=39.1^{\circ}$. According to the law of cosines $d_2^{2}=d_1^{2}+a^{2}-2d_1a\cos\beta$. Here we assume the initial position of the ship relative to the lighthouse and the new position form a triangle. If we consider the right - angled triangle formed by the change in the ship's position, we can also use trigonometry. Let's use the law of cosines. But if we consider the right - triangle formed by the new situation, we know that we can use the tangent function in a right - triangle formed by the components of the ship's movement. However, using the law of cosines:
Let the initial position of the ship and the lighthouse and the new position of the ship form a triangle. We know one side $a = 3.2$ km and the included angle $\beta = 39.1^{\circ}$. Assume the initial distance from the lighthouse to the ship is $x$ (not given, but we can use the fact that we can consider the right - triangle formed by the new position). Let's use the right - triangle approach. If we consider the right - triangle with the angle of $39.1^{\circ}$ and the side of length $3.2$ km. We want to find the hypotenuse $d$. We know that $\sin39.1^{\circ}=\frac{3.2}{d}$.

Step 2: Solve for $d$

$d=\frac{3.2}{\sin39.1^{\circ}}$. Since $\sin39.1^{\circ}\approx0.6293$, $d=\frac{3.2}{0.6293}\approx5.1$ km.

Answer:

1. a. 5.1 km
2. Approximately $73.4^{\circ}$