Question

nearby buildings a and b are built on the same horizontal plane. from point p on the closest edge of the top of building a to building b, an observer finds that the closest point q of the top of building b has an angle of elevation of 44 degrees and 40 minutes. also from point p, the observer finds that the point on the ground that is directly below point q has an angle of depression of 34 degrees and 30 minutes. If building a is known to be 123 feet tall, what is the distance of the observer from point q?

Answer 1

Let's break down the information given step by step:

1. Angle of Elevation from Point P to Point Q (on building B): 44 degrees and 40 minutes.

2. Angle of Depression from Point P to the Ground (directly below Point Q): 34 degrees and 30 minutes.

3. Height of Building A (known): 123 feet.

We need to find the distance of the observer from point Q. Let's denote this distance as "d."

First, let's work with the angle of depression to find the height of point Q above the ground:

1. Height above ground = Height of building A + Height of point Q above building A

2. Height above ground = 123 feet + d * tan(34 degrees + 30 minutes)

Next, we'll work with the angle of elevation to find the height of point Q above the ground:

1. Height above ground = Height of point Q * tan(44 degrees + 40 minutes)

Since both expressions equal the height above the ground, we can set them equal to each other:

123 + d * tan(34.5°) = Height of point Q * tan(44.667°)

Now, we can solve for the height of point Q above building A:

Height of point Q = (123 + d * tan(34.5°)) / tan(44.667°)

Substitute the known values (degrees should be converted to radians) and solve for the height of point Q.

Now, to find the distance "d" of the observer from point Q, we can use trigonometry based on the angle of elevation:

tan(44.667°) = Height of point Q / d

Solve for "d" by rearranging the equation:

d = Height of point Q / tan(44.667°)

Substitute the calculated value for the height of point Q and solve for "d."

Please note that angles should be converted to radians when using trigonometric functions. Also, ensure consistent units (e.g., degrees or radians) when performing calculations.