Question

1. you are building a ramp that has an elevation of 24 in. the distance covered by the ramp will be 48 inches. what is the angle of elevation? round to the nearest tenth.
2. you are surveying a building 50 ft away. the angle of elevation is 45° to the top of the building. what is the height of the building?

Explanation:

3.

Step1: Recall the tangent - angle relationship

The tangent of the angle of elevation $\theta$ in a right - triangle is given by $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the vertical distance (opposite side) is $y = 36$ inches and the horizontal distance (adjacent side) is $x = 75$ inches.
$\tan\theta=\frac{y}{x}$

Step2: Substitute the values

Substitute $y = 36$ and $x = 75$ into the formula: $\tan\theta=\frac{36}{75}=0.48$

Step3: Find the angle

To find the angle $\theta$, we take the inverse tangent (arctan) of 0.48. $\theta=\arctan(0.48)$
Using a calculator, $\theta\approx25.6^{\circ}$

4.

Step1: Recall the tangent - angle relationship

In a right - triangle formed by the observer, the base of the building, and the line of sight to the top of the building, if the distance from the observer to the building (adjacent side) is $x = 50$ ft and the angle of elevation is $\theta=45^{\circ}$, and the height of the building is $h$ (opposite side). We know that $\tan\theta=\frac{h}{x}$.

Step2: Substitute the values

Since $\tan45^{\circ}=1$ and $x = 50$ ft, substituting into the formula $\tan\theta=\frac{h}{x}$, we get $1=\frac{h}{50}$.

Step3: Solve for the height

Multiply both sides of the equation by 50: $h = 50\times1=50$ ft

Answer:

1. The angle of elevation is approximately $25.6^{\circ}$.
2. The height of the building is 50 ft.