Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0017 \mathrm {~cm}$ thick to a hemispherical dome with diameter $70 \mathrm {~m}$.
A) $4.165 \pi$
B) $2.52 \pi$
C) $4.11 \pi$
D) $3.82 \pi$
E) $2.28 \pi$

Question

Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0017 \mathrm {~cm}$ thick to a hemispherical dome with diameter $70 \mathrm {~m}$. 
A) $4.165 \pi$
B) $2.52 \pi$
C) $4.11 \pi$
D) $3.82 \pi$
E) $2.28 \pi$

Quizplus · Accepted Answer

The surface area $A$ of a hemisphere is given by $2\pi r^2$, where $r$ is the radius of the hemisphere. For a hemispherical dome with a diameter of $70 \mathrm {~m}$, the radius $r$ is $35 \mathrm {~m}$. The differential $dA$ gives the change in surface area for a small change in radius $dr$, which in this case corresponds to the thickness of the paint layer. Using the formula for the derivative of the surface area, $dA = 4\pi r dr$, and substituting $r = 35 \mathrm {~m}$ and $dr = 0.0017 \mathrm {~cm} = 0.000017 \mathrm {~m}$, we get $dA = 4\pi (35)(0.000017) = 4.165 \pi \mathrm {~m^2}$. This is the estimated amount of paint needed.

Use Differentials to Estimate the Amount of Paint Needed to Apply