Dome Height

[greatwhiteshark]

On a perfectly circular, small, flat island, a glass dome covers the island, forming a perfect hemisphere. If the canopy is 10 feet tall at the center of the island, how far from the center can 6-foot-tall Omar walk upright?

 

[soroban]

Hello, greatwhiteshark!

Did you make a sketch?

On a perfectly circular, small, flat island, a glass dome covers the island, forming a perfect hemisphere.
If the canopy is 10 feet tall at the center of the island,
how far from the center can 6-foot-tall Omar walk upright?

       * *
       |    *
       |      *
     10|     /| * 
       | 10/  |6 *
       | /    |
   - - + - - -+- - 
           x

We have a right triangle with one side 6 and hypotenuse 10.
Can you find x, the third side?

 

[greatwhiteshark]

yes

Yes, I can find x.

 

[LordExile]

How the height of the dome affects the radius?

So I know the original dome will be 1500 feet tall, and have a radius of 318 feet. I want to know how the radius will be altered if I change the height to 40 feet tall in order for the dome to retain its original surface area?