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Problem 36 Medium Difficulty
Use differentials to estimate the amount of paint needed to apply a coat of paint $ 0.05 cm $ thick to a hemispherical dome with diameter $ 50 m. $
Answer
$d V=2 \pi(25 m)^{2}(0.0005 m)=1.96 m^{3}$
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Video Transcript
So we want to know, Um, how much paint Approximately would be needed to apply a coat to a hemispherical dome. So we know that the volume of the hemispherical dome is one half of the volume of us here. So if the volume of a sphere is four thirds pi r cubed, we know that the volume we're focusing on is two thirds pi r cubed. Then taking the derivative of this, we get that devi d r is equal to two pi r squared and then putting the derivative into differential form. We get that DV equals two pi r squared d r. Now the reason why we want to do this is because we know that are is the original radius before paint is applied, d r is similar to delta are or the change in the radius which we know is going to be 0.5 Then when we plug in these values, we get that DV equals two pi 25 meters time, 0.0 05 m because we converted it from centimeters which is equal to 1.96 m cubed eso that right there would be the volume of paint or the amount of paint needed to cover the hemispherical dome
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