Problem 82
Question
The work \(W\) (in foot-pounds) done in compressing a volume of 9 cubic feet at a pressure of 15 pounds per square inch to a volume of 3 cubic feet is \(W=19,440(\ln 9-\ln 3)\). Find \(W\)
Step-by-Step Solution
Verified Answer
After performing the calculations, the work \(W\) comes out to be approximately 20054.4 foot-pounds.
1Step 1: Analyze the given expression
The given work formula is \(W=19,440(\ln 9-\ln 3)\). This can be rewritten using the property of logarithms \( \ln a - \ln b = \ln \left(\frac{a}{b}\right)\).
2Step 2: Apply the property of logarithms
Substitute \(\ln9 - \ln3\) with \(\ln\left(\frac{9}{3}\right)\) in the work formula. This simplifies the formula to \(W = 19440 \cdot \ln\left( \frac{9}{3} \right)\)
3Step 3: Calculate the result
Now it's a matter of calculation. Evaluate \(\frac{9}{3}\) which equals 3. compute \(\ln3\) and multiply the result by 19440 to get the value of \(W\).
Key Concepts
Properties of LogarithmsNatural Logarithm CalculationApplied MathematicsWork and Energy in Physics
Properties of Logarithms
Understanding the properties of logarithms is crucial for simplifying complex logarithmic expressions. Logarithms, the inverse operation to exponentiation, have unique properties that allow us to manipulate and combine them algebraically.
One fundamental property is the difference of logarithms, which states that \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \) for any positive numbers \(a\) and \(b\). This property reflects the division of the corresponding exponential values and is pivotal in simplifying the expression given in our exercise. Another vital property is the product of logarithms, which expresses that \( \ln a + \ln b = \ln(ab) \), showing that the sum of two logs is the log of the product of their values.
These properties enhance our ability to work with logarithmic equations, enabling us to solve a broad range of problems in mathematics and its applications.
One fundamental property is the difference of logarithms, which states that \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \) for any positive numbers \(a\) and \(b\). This property reflects the division of the corresponding exponential values and is pivotal in simplifying the expression given in our exercise. Another vital property is the product of logarithms, which expresses that \( \ln a + \ln b = \ln(ab) \), showing that the sum of two logs is the log of the product of their values.
These properties enhance our ability to work with logarithmic equations, enabling us to solve a broad range of problems in mathematics and its applications.
Natural Logarithm Calculation
The natural logarithm calculation involves using the logarithmic function with base \(e\), where \(e\) is the irrational constant approximately equal to 2.71828. The notation for the natural logarithm of a number \(x\) is \(\ln x\).
To calculate the natural logarithm of a number, we usually use a scientific calculator or mathematical software, as it involves a non-trivial computation process. For example, \(\ln 3\) refers to the power to which we must raise \(e\) to get 3. It is an indispensable tool in various fields, including science, engineering, and economics, where growth processes are modelled or time-value of money is calculated.
To calculate the natural logarithm of a number, we usually use a scientific calculator or mathematical software, as it involves a non-trivial computation process. For example, \(\ln 3\) refers to the power to which we must raise \(e\) to get 3. It is an indispensable tool in various fields, including science, engineering, and economics, where growth processes are modelled or time-value of money is calculated.
Applied Mathematics
Applied mathematics is the branch of mathematics that deals with mathematical methods used in practical applications. It involves using advanced mathematical techniques to solve complex problems in the real world, including in fields such as engineering, physics, economics, and beyond.
In our textbook exercise, applied mathematics is demonstrated through the use of logarithmic functions to calculate work done in a physical process. More generally, applied mathematics encompasses a broad array of numerical methods, algorithms, and analytical theories to address problems ranging from predicting economic trends to designing efficient structures and resolving scientific phenomena.
In our textbook exercise, applied mathematics is demonstrated through the use of logarithmic functions to calculate work done in a physical process. More generally, applied mathematics encompasses a broad array of numerical methods, algorithms, and analytical theories to address problems ranging from predicting economic trends to designing efficient structures and resolving scientific phenomena.
Work and Energy in Physics
The concept of work and energy in physics plays a fundamental role in understanding how forces cause objects to move or change. The work done by a force is the energy transferred when an object is moved over a distance by that force. It is calculated as the product of the force applied and the distance over which it is applied, in the direction of the force.
In the case of compressing a gas, work is done against the gas pressure to reduce its volume, as seen in the textbook example. The formula \(W=19,440(\ln 9-\ln 3)\) is derived from the principles that govern the relationship between pressure, volume, and energy in a thermodynamic context. Through algebraic operations and the properties of logarithms, such relationships enable us to precisely quantify work in various physical processes.
In the case of compressing a gas, work is done against the gas pressure to reduce its volume, as seen in the textbook example. The formula \(W=19,440(\ln 9-\ln 3)\) is derived from the principles that govern the relationship between pressure, volume, and energy in a thermodynamic context. Through algebraic operations and the properties of logarithms, such relationships enable us to precisely quantify work in various physical processes.
Other exercises in this chapter
Problem 82
Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln (x+1)-\ln (x-2)=\ln x$$\ln (x+1)-\ln (x-2)=\ln x\)
View solution Problem 82
Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{9} \
View solution Problem 83
Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{2}(2 x-3)=\log _{2}(x+4)\)
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Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt[3
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