A balloon, which always remains spherical, has a variable diameter (3)/(2)(2 x+1). Find the rate

step 1 In the question, it is given a ballon which always remains spherical has variable diameter 3x2 i

A balloon, which always remains spherical, has a variable...

Key Concepts

Volume of a Sphere

This concept refers to the formula V = 4/3 ? r³, which gives the volume of a sphere in terms of its radius. It is a fundamental geometric formula used in problems involving spherical shapes and helps establish the relationship between the sphere's radius and its volume.

Chain Rule

The chain rule is a critical tool in calculus for differentiating composite functions. When a variable depends on another variable through one or more intermediary functions, the chain rule allows us to differentiate the outer function with respect to the inner function, and then multiply by the derivative of the inner function.

Variable Substitution

Variable substitution is the process of expressing one variable in terms of another to simplify the differentiation process. In many problems, quantities such as radius or diameter are expressed as functions of an independent variable, and substituting one into a function like the volume helps in applying the chain rule effectively.

Rate of Change

The rate of change measures how one quantity changes in response to changes in another quantity. In calculus, it is represented by the derivative, which provides insights into how fast a function's output is changing concerning its input. This concept is essential when determining how a physical quantity evolves over time or with respect to other variables.

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