From the top of a 200 metres high building, the angle of depression to the bottom of a second building is 20°. From the same point, the angle of elevation to the top of the second building is 10°. Calculate the height of the second building.

The distance z between the buildings can be found via

200/z = tan 20°

Now, the height you want can be found using
h = 200 + z tan 10°

To solve this problem, we can use trigonometric ratios such as tangent.

Let's label the height of the second building as 'h'.

1. First, we need to find the distance from the top of the first building to the bottom of the second building. This can be calculated by using the tangent of the angle of depression:

tan(20°) = height of second building / distance between the two buildings

tan(20°) = h / x (let x be the distance between the two buildings)

2. Next, we need to find the distance from the top of the first building to the top of the second building. This can be calculated by using the tangent of the angle of elevation:

tan(10°) = (h + 200) / x

tan(10°) = (h + 200) / x

3. Rearrange the equations to solve for h:

tan(20°) = h / x --> 1)
tan(10°) = (h + 200) / x --> 2)

From equation (1), we can write: h = x * tan(20°)

Substitute h in equation (2):

tan(10°) = (x * tan(20°) + 200) / x

4. Solve for x:

Multiply both sides of the equation by x:

x * tan(10°) = x * tan(20°) + 200

x * tan(10°) - x * tan(20°) = 200

x * (tan(10°) - tan(20°)) = 200

Divide both sides of the equation by (tan(10°) - tan(20°)):

x = 200 / (tan(10°) - tan(20°))

5. Substitute the value of x back into equation (1) to find h:

h = x * tan(20°)

h = [200 / (tan(10°) - tan(20°))] * tan(20°)

Calculate this value to find the height of the second building.

To find the height of the second building, we can use trigonometry.

Let's label the height of the second building as h.

From the given information, we know that the angle of depression from the top of the first building to the bottom of the second building is 20°. This means that the line of sight is downward at an angle of 20° from the horizontal.

Similarly, the angle of elevation from the same point to the top of the second building is 10°. This means that the line of sight is upward at an angle of 10° from the horizontal.

We can use the tangent function to relate the angles with the height of the second building.

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

For the angle of depression:

tangent(20°) = h / 200

For the angle of elevation:

tangent(10°) = h / 200

Now, let's solve these two equations to find the value of h.

First, rearrange the equation for the angle of depression:

h = 200 * tangent(20°)

Next, rearrange the equation for the angle of elevation:

h = 200 * tangent(10°)

Using a scientific calculator or trigonometric tables, find the values of tangent(20°) and tangent(10°).

Plug in these values into the equations and calculate h.