Textbook Question
5–8. Calculate dy/dx using implicit differentiation.
sin y+2 = x
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.11.19
A spherical balloon is inflated and its volume increases at a rate of 15 in³/min. What is the rate of change of its radius when the radius is 10 in?
Verified step by step guidance1
Start by recalling the formula for the volume of a sphere: V = (4/3)πr³, where V is the volume and r is the radius.
Differentiate the volume formula with respect to time t to find the relationship between the rate of change of volume and the rate of change of radius. This gives us: dV/dt = 4πr²(dr/dt).
Substitute the given rate of change of volume, dV/dt = 15 in³/min, into the differentiated equation: 15 = 4π(10)²(dr/dt).
Solve for dr/dt, the rate of change of the radius, by isolating dr/dt in the equation: dr/dt = 15 / (4π(10)²).
Simplify the expression to find the rate of change of the radius when the radius is 10 inches. This will give you the value of dr/dt in inches per minute.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of a Sphere
The volume of a sphere is given by the formula V = (4/3)πr³, where V is the volume and r is the radius. This relationship shows how the volume changes with respect to the radius, which is crucial for understanding how the volume increase affects the radius.
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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 relate the rate of change of the volume of the balloon to the rate of change of its radius, using derivatives to connect these rates.
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Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. In this context, it allows us to express the derivative of the volume with respect to time in terms of the derivative of the radius with respect to time, facilitating the calculation of the radius's rate of change.
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Related Practice
Textbook Question
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Textbook Question
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.
Textbook Question
Evaluate the derivative of the following functions.
f(y) = tan-1 (2y2 - 4)
Textbook Question
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Textbook Question
Derivatives Find and simplify the derivative of the following functions.
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