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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. $
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00:51
Amrita Bhasin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 10
Linear Approximation and Differentials
Derivatives
Differentiation
Morgan L.
June 28, 2022
The question is to apply a coat of paint but not to fill the dome with paint. Both answers use Volume instead of surface area to reach the answer, which is clearly wrong.
Missouri State University
Campbell University
Baylor University
University of Michigan - Ann Arbor
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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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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