Question

An aerial photographer who photographs real estate properties has determined that the best photo is taken at a height of approximately 456 ft and a distance of 821 ft from the building. What is the angle of depression from the plane to the building

Answer 1

The angle of depression from the plane to the building is approximately 27.4 degrees.

Explanation:

To find the angle of depression, we need to draw a right triangle with the building, the plane, and the ground as its three vertices.

Let's call the height of the plane above the ground h, and let's call the angle of depression we're looking for θ.

Then, we can use trigonometry to solve for θ.

First, we can find the length of the adjacent side of the triangle (the distance from the building to the plane) using the formula for the tangent of an angle:

tan(θ) = opposite / adjacent

tan(θ) = h / 821

We can rearrange this equation to solve for h:

h = 821 * tan(θ)

Next, we know that the height of the plane above the ground is 456 ft, so we can use the Pythagorean theorem to find the length of the hypotenuse of the triangle (the distance from the plane to the ground):

h^2 + 821^2 = c^2

456^2 + 821^2 = c^2

c = sqrt(456^2 + 821^2)

Finally, we can use the formula for the sine of an angle to solve for θ:

sin(θ) = opposite / hypotenuse

sin(θ) = h / c

sin(θ) = (821 * tan(θ)) / sqrt(456^2 + 821^2)

We can solve for θ by taking the inverse sine of both sides:

θ = sin^-1((821 * tan(θ)) / sqrt(456^2 + 821^2))

Unfortunately, this equation cannot be solved algebraically, so we must use a numerical method such as trial and error or a graphing calculator.

Using trial and error or a graphing calculator, we find that θ is approximately 27.4 degrees.

Therefore, the angle of depression from the plane to the building is approximately 27.4 degrees.