Question

question
from a hot - air balloon, genesis measures a $40^{circ}$ angle of depression to a landmark thats 799 feet away, measuring horizontally. whats the balloons vertical distance above the ground? round your answer to the nearest hundredth of a foot if necessary.
answer
attempt 1 out of 2
feet

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the horizontal distance (adjacent side) is 799 feet, the vertical distance (opposite side) is \( x \) (what we need to find), and the angle of depression is \( 40^\circ \). The angle of depression is equal to the angle of elevation from the landmark to the balloon, so we can use the tangent function: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \).

Step2: Set up the equation

Here, \( \theta = 40^\circ \), opposite side is \( x \), and adjacent side is 799. So \( \tan(40^\circ)=\frac{x}{799} \).

Step3: Solve for \( x \)

Multiply both sides by 799: \( x = 799\times\tan(40^\circ) \). Using a calculator, \( \tan(40^\circ)\approx0.8391 \). Then \( x\approx799\times0.8391 \).

Step4: Calculate the value

\( 799\times0.8391\approx670.44 \) (rounded to the nearest hundredth).

Answer:

\( 670.44 \)