The angle of elevation and angle of depression of the top and base of a mobile tower from a mobile handset are 60 degree and 30 degree respectively. If the distances of the top and the base of the tower from the mobile handset are 15m and 8m respectively, find the height of the tower.

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To find the height of the tower, we can use the trigonometric ratios of sine and tangent.

Let's consider a right triangle ABC, where A is the mobile handset, B is the top of the tower, and C is the base of the tower. Here, the angle of elevation is the angle of the triangle at B (top of the tower) and the angle of depression is the angle of the triangle at C (base of the tower).

We are given that the angle of elevation is 60 degrees and the angle of depression is 30 degrees. We are also given the distances between the mobile handset and both the top and base of the tower, which are 15m and 8m respectively.

Step 1: Find the height of the triangle BC (the vertical distance from the base of the tower to the top of the tower).

In triangle ABC, we will use the tangent function:
tan(angle of elevation) = height of the tower / distance between the mobile handset and the top of the tower
tan(60 degrees) = BC / 15m

Rearranging the equation, we can find the height of the tower:
BC = tan(60 degrees) * 15m

Step 2: Find the length of the base of the triangle BC.

In triangle ABC, we can use the tangent function again:
tan(angle of depression) = BC / distance between the mobile handset and the base of the tower
tan(30 degrees) = BC / 8m

Solving for BC, we get:
BC = tan(30 degrees) * 8m

Now we have found the values of BC from both equations, we can compare them.

Since BC is the same for both calculations, we have:
tan(60 degrees) * 15m = tan(30 degrees) * 8m

Now we can solve this equation to find the value of BC.

tan(60 degrees) = (√3) / 1
tan(30 degrees) = 1 / (√3)

Substituting these values into the equation:
(√3) / 1 * 15m = 1 / (√3) * 8m

Simplifying further, we get:
15 * √3 = 8

Dividing both sides of the equation by √3, we find:
15 = 8 / √3

Now, rationalize the denominator by multiplying numerator and denominator by √3:
15 = 8√3 / 3

Finally, solve for the height of the tower (BC):
BC = 8√3 / 3

Therefore, the height of the tower is approximately 4.62 meters.