A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then the height of balloon is given by

Let's denote the angular elevations at points A, B, and C as α, 2α, and 3α respectively. Also, let's denote the height of the balloon as h.

From the problem, we know that the points A, B, and C are on a straight line, with B between A and C. The distances between A and B, and B and C are given as 200m and 100m respectively. Therefore, the distance between A and C is 300m.

We can use the tangent of the angles of elevation to find the height of the balloon. The tangent of an angle in a right triangle is the ratio of the opposite side (in this case, the height of the balloon) to the adjacent side (in this case, the horizontal distance from the point of observation to the point directly beneath the balloon).

So, we have:

tan(α) = h / AB tan(2α) = h / BC tan(3α) = h / AC

Substituting the given distances into these equations, we get:

tan(α) = h / 200 tan(2α) = h / 100 tan(3α) = h / 300

We can solve these equations simultaneously to find the value of h. However, since we have three equations with only one unknown, we can use any two of them to find the value of h.

Let's use the first and second equations:

tan(α) = h / 200 tan(2α) = h / 100

Dividing the second equation by the first, we get:

tan(2α) / tan(α) = (h / 100) / (h / 200)

Simplifying, we get:

2 = 200 / 100

Therefore, h = 100m.

So, the height of the balloon is 100 metres.

Let's denote the angular elevations at points A, B, and C as α, β, and γ respectively. According to the problem, β = 2α and γ = 3α.

Let's denote the height of the balloon as h.

From point A, the tangent of the angle of elevation is equal to the height of the balloon divided by the distance from the point to the balloon. Therefore, we have:

tan(α) = h / d1

where d1 is the distance from point A to the balloon.

Similarly, from point B, we have:

tan(β) = h / d2

where d2 is the distance from point B to the balloon.

Since β = 2α, we can substitute β into the second equation to get:

tan(2α) = h / (d1 + 200)

Since tan(2α) = 2tan(α) / (1 - tan²(α)), we can substitute this into the equation to get:

2tan(α) / (1 - tan²(α)) = h / (d1 + 200)

Solving this equation for h, we get:

h = 2tan(α) * (d1 + 200) / (1 - tan²(α))

Similarly, from point C, we have:

tan(γ) = h / d3

where d3 is the distance from point C to the balloon.

Since γ = 3α, we can substitute γ into the third equation to get:

tan(3α) = h / (d1 + 200 + 100)

Since tan(3α) = 3tan(α) - 4tan³(α) / (1 - 3tan²(α)), we can substitute this into the equation to get:

3tan(α) - 4tan³(α) / (1 - 3tan²(α)) = h / (d1 + 300)

Solving this equation for h, we get:

h = (3tan(α) - 4tan³(α)) * (d1 + 300) / (1 - 3tan²(α))

By equating the two expressions for h, we can solve for α, and then substitute α back into either expression for h to find the height of the balloon.