Question

1. find the measure of the angle of elevation of the sun when a boy 5 feet tall casts a shadow 5 feet long.
2. find the measure of the angle of elevation of the sun when a vertical post 15 feet tall casts a shadow 20 feet long.
3. after takeoff, a plane flies in a straight line for a distance of 4000 feet in order to gain an altitude of 800 feet. find the angle of elevation from the ground to the plane.
4. a 40 foot ladder which is leaning against a wall reaches the wall at a point 36 feet above the ground. find the measure of the angle created between the ladder and the ground.

Explanation:

7.

Step1: Set up tangent ratio

The tangent of the angle of elevation $\theta$ is the ratio of the height (opposite side) to the length of the shadow (adjacent side). So $\tan\theta=\frac{5}{5} = 1$.

Step2: Find the angle

We know that if $\tan\theta = 1$, then $\theta=\arctan(1)$. Since $\arctan(1)=45^{\circ}$, the angle of elevation of the sun is $45^{\circ}$.

Step1: Set up tangent ratio

Let the angle of elevation be $\theta$. Then $\tan\theta=\frac{15}{20}=\frac{3}{4}= 0.75$.

Step2: Find the angle

$\theta=\arctan(0.75)$. Using a calculator, $\theta\approx36.87^{\circ}$.

Step1: Set up tangent ratio

Let the angle of elevation be $\theta$. The plane's altitude is the opposite - side and the distance it flies is the adjacent - side. So $\tan\theta=\frac{800}{4000}=0.2$.

Step2: Find the angle

$\theta = \arctan(0.2)$. Using a calculator, $\theta\approx11.31^{\circ}$.

Answer:

$45^{\circ}$

8.