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Chapter 0: Problem 83

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

Short Answer

Expert verified
The inverse function is \(y = \sqrt{(x - 245.50) / 0.03}\) where \(x\) represents exhaust temperature and \(y\) represents the percent load. After graphing the inverse function, it is seen that for the temperature not to exceed 500 degrees Fahrenheit, the percent load range calculated from the inverse function.

Step by step solution

01

Find the Inverse Function

To find the inverse, we need to switch the positions of \(y\) and \(x\) and solve for \(y\).Starting with the given function: \(y = 0.03x^2 + 245.50\).Interchanging \(x\) and \(y\), we get: \(x = 0.03y^2 + 245.50\).Solving for \(y\), we subtract 245.50 from both sides: \(x - 245.50 = 0.03y^2\).Then we divide by 0.03, take square root: \(y = \sqrt{(x - 245.50) / 0.03}\).In the inverse function, \(x\) now represents the exhaust temperature in degrees Fahrenheit and \(y\) the corresponding percent load on the diesel engine.
02

Graph the Inverse Function

Using any graphing tool or calculator, plot the inverse function \(y = \sqrt{(x - 245.50) / 0.03}\). The result is a square root graph shifted 245.50 units to the right.
03

Find the Percent Load Interval

Considering that the exhaust temperature does not exceed 500 degrees Fahrenheit, the percent load interval can be found by substituting \(x = 500\) in the inverse function: \(y = \sqrt{(500 - 245.50) / 0.03}\). By calculating the above expression, the percent load interval is obtained.

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Key Concepts

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

Exhaust Temperature Function
Understanding the exhaust temperature function is critical for monitoring the efficiency and safety of diesel engines. In our exercise, the function y=0.03x^2+245.50, where y stands for the exhaust temperature in degrees Fahrenheit and x represents the percent load on the diesel engine, models the relationship between the engine load and the exhaust temperature.

For engineers and mechanics, this function is essential. It allows them to predict how temperature will change as the load on the engine increases. With the given restriction of 0, it's clear the function is only relevant within the operational range of the engine load.

Practically, if engineers determine that the exhaust temperature is too high for a given load, they might infer there could be issues with the engine's performance or cooling system. On the other hand, an unexpectedly low temperature could indicate incomplete combustion or other inefficiencies.
Percent Load Diesel Engine
When talking about the 'percent load' of a diesel engine, we're referring to the level of demand placed on the engine compared to its maximum output. In engineering terms, this is an essential concept, as operating a diesel engine at different loads influences fuel efficiency, emission rates, and the engine's lifecycle.

The percent load can be influenced by factors such as the weight the vehicle is carrying or the speed at which it's traveling. For instance, an engine running at a high percent load is under more stress and typically generates more heat 鈥� hence the significance of understanding and monitoring the exhaust temperature function.

The inverse function we derived earlier represents how you can calculate the percent load of the engine, given a specific exhaust temperature. It essentially reverses our initial function, giving users the ability to monitor and adjust engine loads to maintain optimal temperatures.
Graphing Inverse Functions
Graphing inverse functions can be a valuable visual tool for understanding the relationship between two interdependent variables. In this case, taking the original function that models the exhaust temperature based on the engine load and flipping it gives us the inverse function.

To graph the inverse relationship, we use a square root function, as shown in the solution (y = \(\sqrt{(x - 245.50) / 0.03}\)). The graph helps visualize the maximum engine load that corresponds to a certain temperature limit. For example, when graphing, you would see a curve that increases as temperature increases, indicating how the percent load affects the temperature.

Students should use a graphing calculator or software to plot this function, watching how the curve shifts. These visual aids complement the computational work by providing an intuitive understanding of how these variables interact. Remember, graphing both functions on the same axes can also demonstrate how one function is the mirror image of the other across the line y=x. This characteristic is fundamental to inverse functions in mathematics.

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Most popular questions from this chapter

In Exercises 39-54, (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 \(f^{-1}\). $$ f(x)=x^{2}, \quad x \geq 0 $$

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ h(x)=|x+4|-|x-4| $$

In Exercises 39-54, (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 \(f^{-1}\). $$ f(x)=3 x+1 $$

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. $$ \text { If } f \text { is an even function, } f^{-1} \text { exists. } $$

In Exercises 69-74, use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ g^{-1} \circ f^{-1} $$
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