Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
- 8. Definite Integrals3h 25m
4. Applications of Derivatives
Related Rates
Problem 7b
Textbook Question
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?
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1
Identify the formula for the volume of a sphere, which is given by V = (4/3)πr³.
Differentiate the volume formula with respect to time t using the chain rule to find dV/dt, which involves dV/dr and dr/dt.
Substitute the expression for dV/dr, which is dV/dr = 4πr², into the differentiated equation to express dV/dt in terms of r and dr/dt.
Plug in the given values: r = 4 inches and dr/dt = 2 in/min into the equation to find dV/dt.
Calculate the result to determine the rate at which the volume is changing at the specified radius.
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