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# Height of a distant skyscraper?

Burj Dubai, the world’s tallest building, can be seen 95 km away. How tall is it?

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### Replies to This Discussion

There is two unknown.

What does "can be seen 95 km away" means ?

We must either know value of θ angle (vision angle) or "perceived height" (h') and perceived distance.

It should then be rephrased in one of these ways :

Burj Dubai, the world’s tallest building, can be seen 95 km away with a sight angle of 30°.

OR

Looking at Burj Dubai, the world’s tallest building, it appear to be at1km and being 8,6m tall.
Knowing that the building is in fact at 95km, how tall is it ?

OR

You could play also on the width ! (but at which point the width isn't enough to see an object at a given distance ?)

In first case, you have the cosine and are searching the sine.
So :
[95000/cos(30)]*sin(30)

In second case, divide the perceptible height by the perceptible distance then multiply the result by the real distance.

Hope it helps.
Thank's for clarifying and attempting the question. The online calculator on this page: Distance to the Horizon Calculator is close to the solution. On the calculator, the observer is at a known height and the distance is unknown. The diagram on that page is relevant. To calculate the height of the skyscraper, we need to switch the position of observer and object, as the observer is on the ground, and the viewed object is the top of the skyscraper. The radius of the Earth is needed for the calculation.
Working backwards, knowing that the height of the tower is 828 m, I calculate that it can be seen from about 102.7 km away (not 95 km). The Distance to Horizon Calculator's answer is consistent with this information. Using your value of 95 km, I got an answer of about 709.5 meters for the height of the tower. However, this answer is nearly 120 m (longer that a football field) too short.