A building is viewed from two points on the ground. From point S, the angle of elevation to the top is 70 degrees. Point T is 30m directly above S. From T, the angle of elevation to the top is 20 degr

Let $h$ be the height of the building (from the ground level at S to the top of the building, B).
Let $x$ be the horizontal distance from point S (and T) to the base of the building.
Let P be the base of the building, so BP = $h$.
Point T is vertically above S, with $ST = 30$ m.
Draw a horizontal line from T to the building, meeting it at point Q.
Then $PQ = ST = 30$ m.
The height from Q to the top of the building B is $BQ = BP - PQ = h - 30$.

From point S, the angle of elevation to the top of the building B is $70^\circ$.
In the right-angled triangle formed by B, P, and S:

This gives us our first equation:

From point T, the angle of elevation to the top of the building B is $20^\circ$.
In the right-angled triangle formed by B, Q, and T:

This gives us our second equation:

From equation (1), we can express $x$ in terms of $h$:

Substitute this expression for $x$ into equation (2):

Rearrange the equation to solve for $h$:

Factor out $h$:

To simplify the term in the parenthesis, find a common denominator:

Now, isolate $h$:

Using a calculator:

Substitute these values into the equation for $h$:

Rounding to two decimal places:

The height of the building is $34.59 \text{ m}$.