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?

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:

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

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

And if:

[tex]H = \frac{a}{sin(\alpha)}[/tex]

Then:

[tex]b = \frac{a}{sin(\alpha)} \sqrt{1 - sin^{2}(\alpha)}\\b = \frac{a}{sin(\alpha)} \sqrt{1 - 1 + cos^{2}(\alpha)}\\[/tex]

[tex]b = \frac{a}{sin(\alpha)} \sqrt{cos^{2}(\alpha)}\\b = a \frac{cos(\alpha)}{sin(\alpha)} = a Cot(\alpha)[/tex]

ajeigbeibraheem ajeigbeibraheem · Answer 2 · 2021-11-15T14:01:26+01:00

The distance between the ground and lowest cloud which denotes the cloud ceiling is 0.273 km

The diagrammatic representation of the question can be seen from the image attached below.

From the image;

we can determine the cloud ceiling by using the tangent of a right-angle triangle.

Recall that:

[tex]\mathbf{tan \theta = \dfrac{opp}{adjacent}}[/tex]

[tex]\mathbf{tan 20^0 = \dfrac{XY}{0.75 \ km }}[/tex]

[tex]\mathbf{0.36397 = \dfrac{XY}{0.75 \ km }}[/tex]

XY = 0.36397 × 0.75 km

XY = 0.273 km

Therefore, we can conclude that the cloud ceiling is 0.273 km away from the ground.

Learn more about the tangent of a right angle here:

https://brainly.com/question/13952108?referrer=searchResults