Problem 108
Question
Circular Wave A marble is dropped into a lake, resulting in a circular wave whose radius increases at a rate of 6 inches per second. Write a formula for \(C\) that gives the circumference of the circular wave in inches after \(t\) seconds.
Step-by-Step Solution
Verified Answer
The formula for the circumference is \( C = 12\pi t \).
1Step 1: Understand the Problem
The problem involves finding a formula for the circumference of a circular wave created by a marble dropped into a lake. The radius of the wave increases at a constant rate of 6 inches per second.
2Step 2: Formula for Circumference of a Circle
Recall the formula for the circumference of a circle, which is given by \( C = 2\pi r \), where \( C \) is the circumference and \( r \) is the radius.
3Step 3: Determine the Radius as a Function of Time
Since the radius \( r \) increases at 6 inches per second, we can express the radius in terms of time \( t \) as \( r = 6t \).
4Step 4: Substitute the Radius into the Circumference Formula
Substitute the expression for the radius from the previous step into the circumference formula: \( C = 2\pi (6t) \).
5Step 5: Simplify the Expression
Simplify the expression obtained in the previous step: \( C = 12\pi t \).
Key Concepts
Circular WaveRadius of a CircleRate of ChangeTime Dependency in Geometry
Circular Wave
A circular wave forms when a disturbance, like a marble dropped into water, creates ripples extending outward uniformly. This wave expands in a circular pattern, where each wave crest maintains the same distance from the initial point of disturbance, creating circles. The wave's contour or path forms the circumference, continually growing with time due to the wave's expanding radius.
Understanding a circular wave's growth helps visualize how the radius and, consequently, the circumference increase over time. Each circle represents a snapshot of time, and with each passing second, the radius expands further, increasing the overall circumference. This relationship provides a dynamic view of geometry in motion, often observed in natural occurrences.
Understanding a circular wave's growth helps visualize how the radius and, consequently, the circumference increase over time. Each circle represents a snapshot of time, and with each passing second, the radius expands further, increasing the overall circumference. This relationship provides a dynamic view of geometry in motion, often observed in natural occurrences.
Radius of a Circle
The radius of a circle is a pivotal measurement representing the distance from the center of a circle to any point along its edge. Mathematically, any changes in the radius directly affect the size of the circle. Given a situation where the radius increases steadily, its measurement transforms effectively into a function of time.
In the case of the expanding circular wave caused by a marble in the water, the radius is not static. Instead, it's defined by the relation \( r = 6t \), where \( r \) is the radius in inches, and \( t \) is time in seconds. This formula succinctly captures how the radius grows linearly over time, allowing us to easily calculate the circumference at any given moment.
In the case of the expanding circular wave caused by a marble in the water, the radius is not static. Instead, it's defined by the relation \( r = 6t \), where \( r \) is the radius in inches, and \( t \) is time in seconds. This formula succinctly captures how the radius grows linearly over time, allowing us to easily calculate the circumference at any given moment.
Rate of Change
A rate of change describes how a quantity varies over time or another independent variable. In our circular wave scenario, the rate at which the radius grows is constant, measured at 6 inches per second. This consistent change enables us to predict how both the radius and circumference will expand as time progresses.
This rate of change is foundational in calculating our wave’s characteristics. It allows us to set up a mathematical model where thirty seconds into the event, the radius reaches 180 inches \( (6 \text{ inches/second} \times 30 \text{ seconds}) \). Understanding the rate of change solidifies our approach to dynamic systems in geometry and science.
This rate of change is foundational in calculating our wave’s characteristics. It allows us to set up a mathematical model where thirty seconds into the event, the radius reaches 180 inches \( (6 \text{ inches/second} \times 30 \text{ seconds}) \). Understanding the rate of change solidifies our approach to dynamic systems in geometry and science.
Time Dependency in Geometry
In many geometric situations, measurements such as radius and circumference evolve as time progresses, introducing a time-dependent component. With time as a factor, a static circle becomes a dynamic entity, growing due to continuous change in its defining parameters.
This time dependency transforms traditional geometry into a field capable of modeling real-life movements, such as expanding wave fronts or a balloon inflating. We observe this in how our circle’s radius formula \( r = 6t \) directly dictates its evolution over seconds. This relationship emphasizes the importance of integrating time into geometric formulas to reflect real-world dynamics accurately.
This time dependency transforms traditional geometry into a field capable of modeling real-life movements, such as expanding wave fronts or a balloon inflating. We observe this in how our circle’s radius formula \( r = 6t \) directly dictates its evolution over seconds. This relationship emphasizes the importance of integrating time into geometric formulas to reflect real-world dynamics accurately.
Other exercises in this chapter
Problem 107
Graph \(f\) and state its domain. $$f(x)=\ln (-x)$$
View solution Problem 108
Graph \(f\) and state its domain. $$f(x)=\ln \left(x^{2}+1\right)$$
View solution Problem 109
Graph \(y=f(x)\). Is \(f\) increasing or decreasing on its domain? $$f(x)=\log _{1 / 2} x$$
View solution Problem 110
Graph \(y=f(x)\). Is \(f\) increasing or decreasing on its domain? $$f(x)=\log _{1 / 3} x$$
View solution