From a window 20m above the ground, a person could see the top of the house across the street at an angle of elevation of 5 degrees. The angle of depression to the base of the house was 15 degrees. What is the height of the house?

I will assume you made a sketch.

On mine, I let the top of the house across the street be P and its bottom Q.
I let the point directly across the window (W) be R
We know QR = 20
tan15° = 20/WR
WR = 20/tan15

In the other triangle,
tan5° = PR/WR
PR = WRtan5 = (20/tan15)(tan5)
= ...

height = 20 + PR
= ...

take over

To solve this problem, we can use basic trigonometry.

Let's denote the height of the house as "h" in meters.

From the window, we have an angle of elevation of 5 degrees. This means that the angle between the line of sight from the person to the top of the house and the horizontal ground is 5 degrees.

We can use the tangent function to calculate the height of the house. The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.

So, for the angle of elevation of 5 degrees:

tangent(5 degrees) = height of the house / distance to the house

We know the distance from the window to the house is the same as the distance from the house to the window, which is the horizontal distance across the street. However, we don't have this information. Therefore, we need to find a way to relate the distance to the height of the window.

To find this relationship, let's consider the angle of depression. The angle of depression to the base of the house is 15 degrees. This means that the angle between the line of sight from the person to the base of the house and the horizontal ground is 15 degrees.

Using the same logic as before, we can use the tangent function to calculate the height of the window from the base of the house:

tangent(15 degrees) = height of the window / distance to the house

As mentioned earlier, the distance to the house is the same in both cases, so we can equate these two equations:

height of the house / distance to the house = height of the window / distance to the house

Simplifying the equation, we get:

height of the house = height of the window

Therefore, the height of the house is equal to the height of the window, which is 20 meters.

To find the height of the house, we can use trigonometric ratios. Let's call the height of the house "h".

First, let's consider the angle of elevation. The person is looking up at an angle of 5 degrees from the horizontal line. This forms a right triangle where the height of the house is the opposite side and the horizontal distance from the window to the house is the adjacent side. The tangent function relates the opposite side and the adjacent side of a right triangle. We can use it to find the height:

tan(5 degrees) = h / x

x represents the horizontal distance from the window to the house. However, we do not know this distance yet. We will come back to it later.

Now, let's consider the angle of depression. The person is looking down at an angle of 15 degrees from the horizontal line. This also forms a right triangle where the height of the house is the opposite side, and the horizontal distance from the window to the house is once again the adjacent side. Again, we can use the tangent function to find the height:

tan(15 degrees) = h / x

Notice that both right triangles share the same horizontal distance, x. Now we can set up a system of equations:

tan(5 degrees) = h / x
tan(15 degrees) = h / x

Solving these equations simultaneously will give us the height of the house, h.

Here's how to solve the system of equations:
1. Divide the first equation by the second equation to eliminate x:
(tan(5 degrees) / tan(15 degrees)) = (h / x) / (h / x)
tan(5 degrees) / tan(15 degrees) = 1
tan(5 degrees) = tan(15 degrees)

2. Use the inverse tangent function to find the angles that have the same tangent value:
5 degrees = 15 degrees

Since 5 degrees is not equal to 15 degrees, there is no solution to this system of equations. This means that the problem is not solvable with the given information.

From a window 20m above the ground, a person could see the top of the house across the street at an angle of elevation of 5 degrees. The angle of depression to the base of the house was 15 degrees. What is the height of the house?