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Chapter 1: Problem 102

The function given by \(y=0.03 x^{2}+245.50, \quad 0

Short Answer

Expert verified
The inverse function of the given function is \(x = \sqrt{\frac{(y - 245.50)}{0.03}}\). By graphing this equation, the behavior of the function can be observed. Additionally, when the engine's exhaust temperature limit (in this case 500 degrees Fahrenheit) is substituted back into this inverse function, the maximum percent load that won't exceed this temperature limit can be calculated.

Step by step solution

01

Find the inverse function

The inverse of a function reverts the output back to its original input. The current function is \(y=0.03 x^{2}+245.50\). To find the inverse, we need to swap \(x\) and \(y\) and then solve for \(y\). After doing a proper algebraic rearrangement, the inverse function becomes \(x = \sqrt{\frac{(y - 245.50)}{0.03}}\). Where \(x\) represents the percent load of a diesel engine and \(y\) represents the exhaust temperature in degrees Fahrenheit.
02

Graph the inverse function

Graphing functions helps visually understand their behavior. We don't need to provide details on how to graph the inverse function here, since you will use a calculator or computer. The function \(x = \sqrt{\frac{(y - 245.50)}{0.03}}\) should be plotted using a proper range that is relevant for your case. From this graph, we can observe the maximum and minimum values for \(y\) (exhaust temperature).
03

Find the percent load interval

We need to limit the temperature to 500 degrees Fahrenheit. Solving the inverse function with \(y = 500\) will give the engine's percent load value that must not be exceeded to avoid surpassing this temperature limit. So, \(x = \sqrt{\frac{(500 - 245.50)}{0.03}}\) should be calculated to find the value of \(x\). This output represents the maximum percent load that the engine can handle without exceeding an exhaust temperature of 500 degrees Fahrenheit. The percent load interval then becomes \(0 < x < x_\text{max}\), where \(x_\text{max}\) is this calculated result.

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Key Concepts

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

Exhaust Temperature
Exhaust temperature in diesel engines refers to the heat level of the gases leaving the engine combustion chamber. This is an important parameter that needs to be monitored as it affects engine efficiency and emissions.
In the function provided, exhaust temperature, denoted as \( y \), is approximated by the formula \( y = 0.03 x^{2} + 245.50 \), where \( x \) is the percent load. This formula indicates that the temperature increases with the square of the load, showing a non-linear relationship.
Understanding exhaust temperature is crucial for maintaining the engine's health and ensuring it operates within safe limits.
Percent Load
Percent load in a diesel engine context refers to how much of the engine's capacity is being utilized.
In our example, \( x \) signifies this load percentage.
  • A percent load of 0 indicates the engine is not being utilized at all.
  • A percent load of 100 means the engine is running at its full capacity.
Understanding percent load helps in optimizing the performance and efficiency of the engine.
Operators often need to balance between fuel efficiency and power output, and this load percentage is a key factor in that balance.
Graphing Functions
Graphing functions is a powerful way to visualize relationships between variables. It allows us to see how changes in one quantity affect another.
In the context of inverse functions, graphing helps us understand how inputs and outputs are interchanged.
For the inverse function \( x = \sqrt{\frac{(y - 245.50)}{0.03}} \), graphing can help visualize how the percent load changes as the exhaust temperature changes.
  • The original graph plots exhaust temperature against percent load.
  • The inverse graph would plot percent load against exhaust temperature.
By interpreting these graphs, you can quickly determine limits and critical points relevant to engine performance.
Temperature Limit
Setting temperature limits helps to ensure the safe operation of diesel engines by preventing overheating. Each engine has a maximum exhaust temperature it can safely handle before sustaining damage or reduced performance.
In this context, the temperature limit is 500 degrees Fahrenheit. When using the inverse function to determine percent load \( x \), we set \( y \) to 500 in the equation \( x = \sqrt{\frac{(500 - 245.50)}{0.03}} \).
Calculating this value will provide the maximum allowable percent load to stay within this temperature limit.
  • Exceeding this load could risk overheating.
  • Maintaining loads within this range ensures efficient and safe functioning.
Understanding and adhering to these temperature limits are crucial for prolonging engine life and maintaining safety.

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Most popular questions from this chapter

Find a mathematical model for the verbal statement. The gravitational attraction \(F\) between two objects of masses \(m_{1}\) and \(m_{2}\) is proportional to the product of the masses and inversely proportional to the square of the distance \(r\) between the objects.

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \hline \end{array} $$

Determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3 / 5} $$

The total numbers of people (in thousands) in the U.S. civilian labor force from 1992 through 2007 are given by the following ordered pairs. $$\begin{array}{ll} (1992,128,105) & (2000,142,583) \\ (1993,129,200) & (2001,143,734) \\ (1994,131,056) & (2002,144,863) \\ (1995,132,304) & (2003,146,510) \\ (1996,133,943) & (2004,147,401) \\ (1997,136,297) & (2005,149,320) \\ (1998,137,673) & (2006,151,428) \\ (1999,139,368) & (2007,153,124) \end{array}$$ A linear model that approximates the data is \(y=1695.9 t+124,320,\) where \(y\) represents the number of employees (in thousands) and \(t=2\) represents 1992 . Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? (Source: U.S. Bureau of Labor Statistics)
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