Problems from The spherical dome: Surface area and volume

A semispherical dome of an observatory of a metallic coat needs to be covered. We have $500 m^{2}$ of metallic coat and the radius of the semisphere is $10 m$ .

If the strategy is to begin covering the dome from above: Will it be possible to cover the totallity of the surface? In the event that we could not: what will be the height of the covered spherical dome?

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Development:

First, we have to check if those $500 m^{2}$ of metal coat are enough to cover the entire dome: $A_{d o m e} = \frac{4 π R^{2}}{2} = 628, 32 m^{2}$ So, the available metallic surface is not enough to cover the dome.

Now, we have to study the height of the covered part of the dome: $A_{d o m e} = 2 π \cdot R \cdot h$ $500 = 2 π \cdot 10 \cdot h$ $h_{d o m e} = 7, 96 m$

Solution:

It is not possible to cover the whole semisphere. The covered part of the dome has a height of $7, 96 m$ .

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