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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 7b
Capitolo 3, Problema 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?

Guida verificata passo dopo passo
1
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.

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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.
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Percorso guidato
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.
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Example 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.
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05:02
Intro to the Chain Rule
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