when viewed from a point P, due north of a tall building, the angle of elevation of the top of the building is 40 degree ; when viewed from a point Q due east of the building, the angle of elevation is 35 degree. If the distance PQ is 120m, find the height of the building.

Let's set up some labels.

T = top of building
B = bottom of building
y = BP
x = BQ
If the building has height h, then

h/x = tan 40°
h/y = tan 35°
so, x tan 40° = y tan 35°
x^2+y^2 = 120^2

solving that, we get
x = 76.88
y = 92.13
h = 64.5

steve you are perfectly crrct. Thank you sir

To find the height of the building, we need to use the concept of trigonometry. Let's analyze the situation step by step.

Step 1: Draw a diagram
Draw a rough diagram with the building, points P and Q, and the angles of elevation.

```
B (Top of the building)
/
/
/|\
/ | \
/ | \
/ | \
P----D----Q
```

Step 2: Identify the given information
From the question, we are given:
- The angle of elevation from point P to the top of the building is 40 degrees.
- The angle of elevation from point Q to the top of the building is 35 degrees.
- The distance PQ is 120 meters.

Step 3: Identify the trigonometric ratios
To solve this problem, we will use the tangent function. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

In right triangle PBD:
- The opposite side is the height of the building (BD).
- The adjacent side is the distance from P to D (PD).

In right triangle QBD:
- The opposite side is the height of the building (BD).
- The adjacent side is the distance from Q to D (QD).

Step 4: Write the equations
Using the tangent function, we can set up two equations:

For triangle PBD:
tan(40 degrees) = BD / PD

For triangle QBD:
tan(35 degrees) = BD / QD

Step 5: Solve the equations
Let's use these equations to find the value of BD.

tan(40 degrees) = BD / PD
BD = tan(40 degrees) * PD -- Equation 1

tan(35 degrees) = BD / QD
BD = tan(35 degrees) * QD -- Equation 2

Step 6: Substitute the values and solve for BD
Substituting the given value PD = 120m into Equation 1 and QD = 120m into Equation 2, we can find the value of BD.

BD = tan(40 degrees) * 120m ≈ 99.999m -- from Equation 1
BD = tan(35 degrees) * 120m ≈ 87.328m -- from Equation 2

Since BD represents the height of the building, the height is approximately 99.999 meters.