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A man at A observes the angle of

elevation of a balloon to be 30 degrees. He then walks 1000

metres towards the balloon to a point B and finds the

elevation to be 60 degrees. If the balloon has height h

metres and the man had x metres left to walk before he is

directly under the balloon show that h=xtan60º and h= (x

+1000) tan30º and use these two equations to find x.

The height of the opposite side of the triangle is the same in both cases and is the height of the balloon. The angles involved are 30º and 60º. The length of the near side is x+1000 and x.

From this, it is known that when the distance is x and the angle is 60º, h/x = tan 60º.

It is also known when the distance is x+1000, the angle is 30º, so we have h/(x+1000)=tan 30º.

Using these two equations, we can solve both for h and get the equations above.

To solve for x, take h = x * tan 60º and put that into the 2nd equation.

This gives (x * tam 60º)/(x+1000) = tan 30º.

To find x, multiply both sides by x+1000, which gets rid of the fraction.

Subtract x * tan 30º from both sides.

Factor the x term out on the left side.

Divide both sides by the multiple of the x term.

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