교과서 질문
At all times, the length of a rectangle is twice the width w of the rectangleas the area of the rectangle changes with respect to time t.
a. Find an equation relating A to w.
386
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
3장, 문제 7b
The volume V of a sphere of radius r changes over time t.
b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?
검증된 단계별 안내1
Start by recalling the formula for the volume of a sphere: V = (4/3)πr^3. This formula gives the volume in terms of the radius r.
To find the rate at which the volume changes with respect to time, we need to differentiate the volume formula with respect to time t. This involves using the chain rule since the radius r is a function of time t.
Apply the chain rule: dV/dt = dV/dr * dr/dt. Here, dV/dr is the derivative of the volume with respect to the radius, and dr/dt is the rate at which the radius changes with respect to time.
Calculate dV/dr by differentiating V = (4/3)πr^3 with respect to r. This gives dV/dr = 4πr^2.
Substitute the given values into the differentiated formula: dr/dt = 2 in/min and r = 4 inches. Plug these into dV/dt = 4πr^2 * dr/dt to find the rate at which the volume is changing.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Related Rates
Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the volume of the sphere changes as the radius changes over time. This requires applying the chain rule of differentiation to relate the rates of change of the radius and the volume.
추천 영상:
가이드 코스
04:16
Intro To Related Rates
Volume of a Sphere
The volume V of a sphere is given by the formula V = (4/3)πr³, where r is the radius. Understanding this formula is crucial because it allows us to express the volume in terms of the radius, which is necessary for calculating how the volume changes as the radius changes.
추천 영상:
08:29Example 5: Packaging Design
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. In this context, we will use the chain rule to differentiate the volume formula with respect to time, allowing us to relate the rate of change of volume to the rate of change of the radius.
추천 영상:
05:02Intro to the Chain Rule
관련 실천
교과서 질문
The volume V of a sphere of radius r changes over time t.
c. At what rate is the radius changing if the volume increases at 10 in³ when the radius is 5 inches?
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Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
f(x) = x(x-1)
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교과서 질문
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
g(t) = (t + 1)(t² - t + 1)
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