1. a) We know the side adjacent to the $50°$ angle and want to find the hypotenuse. We can use the cosine function:

   $$\begin{array}{cccc}\hfill \cos 50°& =\hfill & {\displaystyle \frac{1000\mathrm{ft}}{c}}\hfill & \hfill \\ \hfill c\cos 50°& =\hfill & 1000\mathrm{ft}\hfill & \mathbf{Multiplying\; by}\mathit{c}\hfill \\ \hfill c& =\hfill & {\displaystyle \frac{1000\mathrm{ft}}{\cos 50°}}\hfill & \mathbf{Dividing\; by\; cos}\mathbf{50}°\hfill \\ \hfill & \approx \hfill & 1556\mathrm{ft}.\hfill & \hfill \end{array}$$

   After walking 1000 ft, she is approximately 1556 ft from the entrance to the Cave of the Winds.

2. b) We know the side adjacent to the $50°$ angle and want to find the opposite side. We can use the tangent function:

   $$\begin{array}{ccc}\hfill \tan 50°& =\hfill & {\displaystyle \frac{b}{1000\mathrm{ft}}}\hfill \\ \hfill b& =\hfill & 1000\mathrm{ft}\cdot \tan 50°\approx 1192\mathrm{ft}.\hfill \end{array}$$

   The width of the Niagara River is approximately 1192 ft at that point.

We first find the angle $\theta$ that the rafter makes with the side wall. We know the rise, 10, and the run, 12, so we can use the tangent function to determine the angle that corresponds to the pitch of $10/12$:

Using a calculator, we find that $\theta \approx 39.8°$. Since trigonometric function values of $\theta$ depend only on the measure of the angle and not on the size of the triangle, the angle for the rafter is also $39.8°$.

To determine the length x of the rafter, we can use the cosine function. (See the figure at left.) Note that the width of the house is 42 ft, and a leg of this triangle is half that length, 21 ft.

The length of the rafter for this house is approximately 27.3 ft.

We begin by labeling a drawing with the given information.

1. a) The difference in the elevation of Station St. Sophia and the elevation of the town is $10,550\mathrm{ft}-8725\mathrm{ft}$, or 1825 ft. This measure is the length of the side opposite the angle of elevation, $\theta$, in the right triangle shown at left. Since we know the side opposite $\theta$ and the hypotenuse, we can find $\theta$ by using the sine function. We first find $\theta$:

   

   $$\sin \theta ={\displaystyle \frac{1825\mathrm{ft}}{5750\mathrm{ft}}}\approx \mathrm{0.3174.}$$

   Using a calculator, we find that

   

   Thus the angle of elevation from the town to Station St. Sophia is approximately $18.5°$.

   

2. b) When parallel lines are cut by a transversal, alternate interior angles are equal. Thus the angle of depression, $\beta$. from Station St. Sophia to Mountain Village is equal to the angle of elevation from Mountain Village to Station St. Sophia, so we can use the right triangle shown at left.

   The difference in the elevation of Station St. Sophia and the elevation of Mountain Village is $10,550\mathrm{ft}-9500\mathrm{ft}$, or 1050 ft. Since we know the side opposite the angle of elevation and the hypotenuse, we can again use the sine function:

   $$\sin \beta ={\displaystyle \frac{1050\mathrm{ft}}{3913\mathrm{ft}}}\approx 0.2683.$$

   Using a calculator, we find that

   $$\beta \approx 15.6°.$$

   The angle of depression from Station St. Sophia to Mountain Village is approximately $15.6°$.

From the figure, we have

The height of the bamboo shoot is approximately 74 ft.