The surface area, S, of a spherical balloon of radius r, in...

The surface area, $S$, of a spherical balloon of radius $r$, in inches, is determined by $S(r)=4 \pi r^{2}$. If the balloon is being blown up at a constant rate by a machine, then the radius of the balloon is a function of time. Suppose that this function is $r(t)=1.2 t$, where $t$ is in seconds. a) Determine the radius of the balloon at 2 seconds. b) Determine the surface area at 2 seconds. c) Express the surface area as a function of time by determining $S \circ r$. d) Using the function determined in part c), determine the surface area after 2 seconds. e) Do your answers in parts b) and d) agree? If not, explain why not.

The surface area, $S$, of a spherical balloon of radius $r$, in inches, is determined by $S(r)=4 \pi r^{2}$. If the balloon is being blown up at a constant rate by a machine, then the radius of the balloon is a function of time. Suppose that this function is $r(t)=1.2 t$, where $t$ is in seconds.
a) Determine the radius of the balloon at 2 seconds.
b) Determine the surface area at 2 seconds.
c) Express the surface area as a function of time by determining $S \circ r$.
d) Using the function determined in part c), determine the surface area after 2 seconds.
e) Do your answers in parts b) and d) agree? If not, explain why not.

Key Concepts

Substitution and Evaluation of Functions

This key concept involves replacing the variable in a function with a specific value or with another function. It is the core process used to derive a numerical result or a new function that expresses a dependent relationship, such as substituting a time-dependent radius into the surface area formula to determine the area as a function of time.

Consistency in Mathematical Modeling

Ensuring consistency in mathematical modeling means that results obtained through different valid methods should agree. This concept is important when comparing direct substitution into a formula with results obtained via function composition, reinforcing the idea that different approaches to solving a problem within the same mathematical framework yield consistent and reliable outcomes.

Surface Area of a Sphere

This concept refers to the geometric formula used to calculate the area covering a sphere, expressed as S = 4?r². It is a fundamental part of solid geometry that relates the radius of a sphere to its total surface area, and it is widely applied in various fields including physics, engineering, and everyday problem solving.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In practical terms, when one quantity (such as the radius) is expressed as a function of another variable (like time), this value can then be used as the input for a secondary function (such as the surface area function), thereby linking the two relationships into a composite function that describes how the secondary quantity changes over time.

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