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?
First, we have to check if those $500 \ m^2$ of metal coat are enough to cover the entire dome: $$A_{dome}=\dfrac{4\pi R^2}{2}=628,32 \ m^2$$ 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: $$A_{dome}=2\pi\cdot R \cdot h$$ $$500=2\pi\cdot 10 \cdot h$$ $$h_{dome}=7,96 \ m$$
It is not possible to cover the whole semisphere. The covered part of the dome has a height of $7,96 \ m$.