Question

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.07 cm thick to a hemispherical dome with diameter 60 m. (Round your answer to two decimal places.)

Answer 1

An hemispherical dome is half a sphere. If the diameter is 60 m, then the radius is 30 m.

We can use differentials to solve this problem because we are adding a thin layer to the original dome, so the volume of the dome in increased by a differential of itself.

This differential volume that the dome is increased is equal to the volume of the coat of paint.

The volume of the dome can be written as:

`\begin{gathered} V=(4\pi)/(3)r^3 \\ (dV)/(dr)=(4\pi)/(3)\cdot3r^2\Rightarrow dV=(4\pi)/(3)\cdot3r^2\cdot dr=4\pi r^2\cdot dr \end{gathered}`

Now, we can calculate dV as:

Answer: the paint needed for this coat is approximately 7.92 m^3