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Height of a Dome

Date: 08/06/2002 at 15:20:27
From: Ashley Brannen
Subject: Algebra 2 Word Problems

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

I have tried to solve this problem using radius in some way but I 
am stuck. Any tips?


Date: 08/08/2002 at 13:46:22
From: Doctor Mihir
Subject: Re: Algebra 2 Word Problems

Hello Ashley,

Consider this triangle:

            H
            |\
            | \r
          h |  \
            |   \
            |____\O
              d

Let's say the distance Omar can walk is d ft. and the radius of the 
dome is r, which as we know is 10 ft. We also know that the point 
where Omar touches the dome is on the hemisphere.

We know Omar's height h to be 6 ft.

Now, applying Pythagoras's formula for a right angle triangle:

 r^2 = h^2 + d^2

 d^2 = r^2 - h^2

 d^2 = 10^2 - 6^2

Now, solving for d will give you the distance Omar can travel standing 
upright.

Hope this is helpful. Good luck.

- Doctor Mihir, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Higher-Dimensional Geometry
High School Triangles and Other Polygons

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