Problem 61 - Algebra and Trigonometry

Question

Area of a Ripple \(A\) stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 60 \(\mathrm{cm} / \mathrm{s}\) . (a) Find a function \(g\) that models the radius as a function of time. (b) Find a function \(f\) that models the area of the circle as a function of the radius. (c) Find \(f \circ g .\) What does this function represent?

Step-by-Step Solution

Answer

(a) \( g(t) = 60t \); (b) \( f(r) = \pi r^2 \); (c) \( f \circ g(t) = 3600\pi t^2 \). Function \( f \circ g \) represents the area as a function of time.

Step 1: Model the Radius as a Function of Time

Given that the ripple travels outward at a speed of 60 cm/s, we can express the radius of the ripple as a function of time. Let the radius be denoted by \( r \) and time by \( t \). Since speed is distance divided by time, we have \( r(t) = 60t \).

Step 2: Model the Area as a Function of the Radius

The area \( A \) of a circle is given by the formula \( A = \pi r^2 \). Here, the radius \( r \) is the input for the area function. Therefore, the function \( f \) that models the area in terms of radius is \( f(r) = \pi r^2 \).

Step 3: Find the Composition of Functions

The function \( f \circ g \) is the composition of the functions \( f \) and \( g \). It represents the area of the ripple as a function of time. First, substitute \( g(t) = 60t \) into \( f(r) = \pi r^2 \):\[ f(g(t)) = \pi (60t)^2 = \pi \cdot 3600t^2 = 3600\pi t^2. \]Thus, \( f \circ g = 3600\pi t^2 \).

Key Concepts

Functions of Time and Radius

In mathematical modeling, understanding how different quantities change with time is crucial. Here, we focus on how the radius of a circular ripple changes as time progresses. The given situation tells us that a ripple expands at a constant speed of 60 cm/s. The radius, which we'll denote as \( r \), becomes larger the longer the ripple travels. By expressing the radius as a function of time \( t \), we can model its behavior across any time span. The main idea is that speed is a measure of how quickly something is moving, calculated as the change in distance over time. This relationship can be described by the equation \( r(t) = 60t \).

• \( r(t) \): Radius as a function of time
• \( 60 \): The speed of expansion in cm/s
• \( t \): Time in seconds

This linear equation tells us that for every second that passes, the radius of the ripple increases by 60 cm. Understanding this basic function helps set the groundwork for more complex analyses involving other variables changing with time.

Composition of Functions

The composition of functions is an essential concept in mathematics that reveals how to combine two or more functions to create a new function. This new function melds the outputs and inputs seamlessly to produce meaningful results. In this scenario, we seek to build a function representing the area of a ripple as a function of time, leveraging the radius \( r(t) \) we've established.We start with two functions:

• \( g(t) = 60t \)
• \( f(r) = \pi r^2 \)

The function \( g(t) \) gives us the radius at any time \( t \), while \( f(r) \) computes the area based on the radius. By substituting \( g(t) \) into \( f(r) \), we form the composite function \( f \circ g \). So, \( f(g(t)) = \pi (60t)^2 = 3600\pi t^2 \).

• \( f \circ g \): Composition of \( f \) and \( g \)
• This function expresses the area as a direct function of time

Through composition, we elegantly combine both functions, allowing us to predict how the ripple's area changes over time based solely on its initial condition.

Area of a Circle

The area of a circle is a fundamental geometric concept, providing insight into how much flat space is enclosed by a circle. This concept is especially pertinent when considering circular ripples on a water surface. The mathematical expression for a circle's area is \( A = \pi r^2 \), where \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.In the context of a ripple, understanding area helps us predict how the space covered by the ripple increases over time. With the formula \( f(r) = \pi r^2 \), you substitute the known radius to compute the area. As we've noted, this function \( f(r) \) ties into larger analyses through composition.

• \( A = \pi r^2 \): Formula for area
• \( r \): Radius of the circle

The elegance of the circle area formula lies in its ability to scale with the square of the radius, indicating that small increases in the radius lead to proportionally larger increases in the area. This concept is widely applicable to various scientific and engineering fields, illustrating the power of circular geometry.