The angles of elevation of the top of a tower from two points on the same side of the tower are α and β (α >β). If the distance between the two points is 40m, find the height of the tower.

To solve this problem, we will use the concept of trigonometry. Here are the steps:

1. Let's denote the height of the tower as h and the distance from the tower to the closer observation point as d.

2. From the point of view of the closer observation point, we can form a right triangle with the tower. The tangent of the angle α is equal to the opposite side (which is the height of the tower h) divided by the adjacent side (which is the distance to the tower d). So, we have tan(α) = h/d.

3. From the point of view of the farther observation point, we can form another right triangle with the tower. The tangent of the angle β is equal to the opposite side (which is still the height of the tower h) divided by the sum of the adjacent side and the distance between the two observation points (which is d + 40m). So, we have tan(β) = h/(d + 40).

4. Now we have a system of two equations, and we can solve it for h.

   From the first equation, we can express d as d = h/tan(α).

   Substituting this into the second equation gives us tan(β) = h/(h/tan(α) + 40).

5. Solving this equation for h gives us the height of the tower:

   h = 40*tan(α)*tan(β) / (tan(α) - tan(β)).

So, the height of the tower can be found if the angles α and β are known.