Question

In aviation, it is helpful for pilots to know the cloud ceiling, which is the distance between the ground and lowest cloud. The simplest way to measure this is by using a spotlight to shine a beam of light up at the clouds and measuring the angle between the ground and where the beam hits the clouds. If the spotlight on the ground is 0.75 km from the hangar door. What is the cloud ceiling?

Answer 1

Answer:

$b = a \frac{cos(\alpha)}{sin(\alpha)} = a Cot(\alpha)$

Explanation:

In order to calculate the height reached by the beam when it hit the cloud, it is possible to use the following mathematical equations based on the Pythagoras' theorem:

$b = \sqrt[2]{H^{2} - a^{2}}$

$b = \sqrt[2]{H^{2} - H^{2} sin^{2} (\alpha)} \\b = H \sqrt[2]{1 - sin^{2} (\alpha)}$

And if:

$H = \frac{a}{sin(\alpha)}$

Then:

$b = \frac{a}{sin(\alpha)} \sqrt{1 - sin^{2}(\alpha)}\\b = \frac{a}{sin(\alpha)} \sqrt{1 - 1 + cos^{2}(\alpha)}$

$b = \frac{a}{sin(\alpha)} \sqrt{cos^{2}(\alpha)}\\b = a \frac{cos(\alpha)}{sin(\alpha)} = a Cot(\alpha)$