Question

mountain with a vertical height of 1200 feet. the distance from the top of the mountain to the base is 2400 feet. what is the angle of elevation from the... wants to know the width w of a river to properly set instruments for an experiment. from point a, the biologist walks downstream 100 feet and sights to point t... that θ = 47°.

Explanation:

Looking at the visible part of the problem (about the mountain with vertical height 1200 feet and distance from top to base 2400 feet, finding the angle of elevation), we can solve it using trigonometry (a subfield of Mathematics, specifically Geometry or Trigonometry).

Step 1: Identify the triangle type and trigonometric ratio

We have a right triangle where the vertical height (opposite side to the angle of elevation, let's call the angle \( \theta \)) is \( 1200 \) feet, and the hypotenuse (distance from top to base) is \( 2400 \) feet. The sine function relates the opposite side and the hypotenuse: \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \)

Step 2: Substitute the values

Substitute the opposite side (\( 1200 \)) and hypotenuse (\( 2400 \)) into the sine formula:
\( \sin(\theta)=\frac{1200}{2400} \)
Simplify \( \frac{1200}{2400}=\frac{1}{2} \)

Step 3: Find the angle

We know that \( \sin(\theta)=\frac{1}{2} \), and from the unit circle or trigonometric values, we recall that \( \sin(30^\circ)=\frac{1}{2} \), so \( \theta = 30^\circ \)

Step1: Identify trigonometric ratio

We use the sine function for the right triangle: \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \)

Step2: Substitute values

Substitute opposite \( = 1200 \), hypotenuse \( = 2400 \): \( \sin(\theta)=\frac{1200}{2400}=\frac{1}{2} \)

Step3: Determine the angle

Since \( \sin(30^\circ)=\frac{1}{2} \), the angle \( \theta = 30^\circ \)

Answer:

The angle of elevation is \( 30^\circ \)