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Chapter 13: Problem 28

$$\text {Plot the indicated graphs.}$$ By pumping, the air pressure in a tank is reduced by \(18 \%\) each second. Thus, the pressure \(p\) (in \(\mathrm{kPa}\) ) in the tank is given by \(p=101(0.82)^{t},\) where \(t\) is the time (in s). Plot the graph of \(p\) as a function of \(t\) for \(0 \leq t \leq 30\) s on (a) a regular rectangular coordinate system and (b) a semilogarithmic coordinate system.

Short Answer

Expert verified
Plot \( p = 101(0.82)^t \) for \( 0 \leq t \leq 30 \) on both regular and semilogarithmic graph.

Step by step solution

01

Understand the equation

The equation given is \( p = 101(0.82)^t \), where \( p \) represents pressure in kilopascals and \( t \) represents time in seconds. \( 101 \) is the initial pressure while \( 0.82 \) is the decay factor, representing that the pressure is reduced by \( 18\% \) each second.
02

Create a table of values

Construct a table with values of \( t \) from \( 0 \) to \( 30 \) in steps, e.g., every 5 seconds. For each \( t \), compute \( p \) using the formula \( p = 101(0.82)^t \). This will give you points to plot the graph.
03

Plot on regular coordinate system

Using the values in the table, plot the points \((t, p(t))\) on a regular rectangular coordinate system with the x-axis representing time \( t \) and the y-axis representing pressure \( p \). Connect the points to form a continuous curve showing the decay of pressure over time.
04

Understand semilogarithmic plotting

On a semilogarithmic plot, one of the axes (usually the y-axis) is on a logarithmic scale. This is useful for visualizing exponential decay since it can linearize the curve.
05

Plot on semilogarithmic coordinate system

Use the same values from the table. The x-axis remains linear with \( t \), but plot \( \log(p(t)) \) on the y-axis instead. Plot the points and connect them to form a linear-like trend, indicating exponential decay more clearly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Plotting
Plotting a graph is a visual way to represent data and understand relationships between variables. In this exercise, we examine how to plot the function \( p = 101(0.82)^t \). Here, the variable \( t \) represents time, and \( p \) is the pressure in kilopascals. This function describes an exponential decay, as the pressure decreases over time.When plotting a function:
  • Create a table of values by choosing specific points in time, such as every 5 seconds. This will help you calculate corresponding pressure values using the function.
  • Use these values to plot points on a coordinate system, which helps visually represent how pressure changes over time.
  • Connect the plotted points with a smooth curve to illustrate the continuous nature of exponential decay.
Understanding how to plot these points on a graph helps in visualizing the rate and manner in which pressure decreases with each second.
Semilogarithmic Graph
A semilogarithmic graph uses a logarithmic scale for one axis and a linear scale for the other. This type of graph is particularly useful for depicting exponential relationships, like exponential decay, in a clearer way.Here's why semilogarithmic plots are helpful:
  • They can transform exponential curves into straight lines, making trends easier to interpret.
  • In exponential decay, a semilogarithmic plot helps highlight the rate of change over time more clearly than a standard plot.
When plotting on a semilogarithmic graph:
  • The x-axis is linear, representing time \( t \).
  • The y-axis, however, uses a logarithmic scale to represent \( p \); you plot \( \log(p) \) instead of \( p \).
  • Connecting these points usually results in a linear representation of the original exponential decay function.
This method allows you to see patterns that may not be as obvious on a regular scale, providing a better understanding of how quantities change exponentially.
Rectangular Coordinate System
A rectangular coordinate system, also known as a Cartesian coordinate system, helps plot functions to visually interpret relationships between two variables. For example, plotting function \( p = 101(0.82)^t \) shows how pressure changes over time within a specific domain.In a rectangular coordinate system:
  • The horizontal axis (x-axis) typically represents the independent variable. Here, it's time \( t \), marked in seconds.
  • The vertical axis (y-axis) stands for the dependent variable, which is pressure \( p \) in this context, measured in kilopascals.
  • Each pair of coordinates \((t, p)\) gets plotted as a point on this graph.
  • By connecting these points, the overall trend of the function becomes visible, showing a smooth curve for exponential decay.
This coordinate system facilitates data analysis and helps students visualize mathematical phenomena such as exponential decay linearly over time, illustrating real-world applications, such as the decay of air pressure in a tank.

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