When a pebble is thrown into a pond, the circle formed by...

When a pebble is thrown into a pond, the circle formed by the pebble hitting the water expands with time. The area of the expanding circle may be determined by the formula $A=\pi r^{2}$. The radius, $r$, of the circle, in feet, is a function of time, $t$, in seconds. Suppose that the function is $r(t)=2 t$. a) Determine the radius of the circle at 3 seconds. b) Determine the area of the circle at 3 seconds. c) Express the area as a function of time by determining $A \circ r$. d) Using the function determined in part c), determine the area of the circle at 3 seconds.

When a pebble is thrown into a pond, the circle formed by the pebble hitting the water expands with time. The area of the expanding circle may be determined by the formula $A=\pi r^{2}$. The radius, $r$, of the circle, in feet, is a function of time, $t$, in seconds. Suppose that the function is $r(t)=2 t$.
a) Determine the radius of the circle at 3 seconds.
b) Determine the area of the circle at 3 seconds.
c) Express the area as a function of time by determining $A \circ r$.
d) Using the function determined in part c), determine the area of the circle at 3 seconds.

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Key Concepts

Function Evaluation

This concept involves substituting a specific value into a function to calculate its output. It is essential in problems where you determine the result of a function for a given input, such as calculating the radius at a specific time or evaluating any time-dependent function.

Composite Functions

Composite functions combine two functions where the output of one function becomes the input of another. This is crucial when expressing one quantity as a function of another dependent quantity, such as determining the area of a circle as a function of time by plugging the time-dependent radius into the area formula.

Substitution

Substitution is a technique used to replace a variable or an entire function with an equivalent expression. It is important for rewriting functions in different forms, like replacing the variable in a geometric formula with another function to express a physical quantity in terms of a different independent variable.

Geometric Formulas

Geometric formulas provide relationships among dimensions of geometric figures. The area of a circle, given by the formula A = ?r², is a fundamental concept that combines geometry with algebra, enabling the calculation of areas based on other measurable parameters like the radius.

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