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(a) Use differentials to find a formula for the a…

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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. $


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00:51

Amrita Bhasin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 10

Linear Approximation and Differentials

Related Topics

Derivatives

Differentiation

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ML

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.

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04:40

Derivatives - Intro

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.

Video Thumbnail

44:57

Differentiation Rules - Overview

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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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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Calculus: Early Transcendentals

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Related Topics

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Top Calculus 1 / AB Educators
Catherine Ross

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Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

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.

Video Thumbnail

44:57

Differentiation Rules - Overview

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.

Join Course
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