Problem 51 - Algebra for College Students

Question

The equation \(P(a)=14.7 e^{-0.21 a}\), where \(a\) is the altitude above sea level measured in miles, yields the atmospheric pressure in pounds per square inch. If the atmospheric pressure at Cheyenne, Wyoming, is approximately \(11.53\) pounds per square inch, find that city's altitude above sea level. Express your answer to the nearest hundred feet. 6100 feet

Step-by-Step Solution

Answer

The altitude of Cheyenne, Wyoming, is approximately 6100 feet above sea level.

Step 1: Understand the Problem

We are given an equation for atmospheric pressure as a function of altitude: \(P(a)=14.7 e^{-0.21 a}\). We need to find the altitude \(a\) when the pressure \(P(a)\) is 11.53 psi.

Step 2: Set Up the Equation

Substitute the given pressure into the equation: \(11.53 = 14.7 e^{-0.21 a}\). Now, we need to solve for \(a\) to find the altitude.

Step 3: Isolate the Exponential Term

Divide both sides of the equation by 14.7 to isolate the exponential term: \(\frac{11.53}{14.7} = e^{-0.21 a}\).

Step 4: Solve for the Exponent

Take the natural logarithm (\(\ln\)) of both sides to solve for the exponent: \(\ln\left(\frac{11.53}{14.7}\right) = -0.21 a\).

Step 5: Calculate \(a\)

Rearranging the equation, solve for \(a\): \(a = \frac{\ln\left(\frac{11.53}{14.7}\right)}{-0.21}\). Calculate this using a calculator to find \(a\).

Step 6: Convert Altitude to Feet

The result for \(a\) will be in miles. Multiply by 5280 feet/mile to convert the altitude from miles to feet.

Step 7: Express to the Nearest Hundred Feet

Round the result of the calculation to the nearest hundred feet to match the requirements of the problem.

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. These play a crucial role in situations where quantities grow or decay at rates proportional to their current value. In the problem, the exponential equation is given by \[ P(a) = 14.7 e^{-0.21 a} \]where:- \( P(a) \) represents atmospheric pressure in pounds per square inch (psi),- \( 14.7 \) is the standard atmospheric pressure at sea level,- \( e \) is the base of the natural logarithm,- and the exponent \(-0.21 a\) illustrates how pressure decreases with altitude.To solve the problem, we substitute the given pressure value and solve for the altitude \(a\). The essence of solving such equations is manipulating the exponential term, often requiring logarithms to "undo" the exponential.
This approach allows us to isolate the variable and solve accordingly.

Natural Logarithms

Natural logarithms, denoted as \( \ln \), are logarithms with base \( e \), where \( e \approx 2.718 \). They are instrumental in solving exponential equations because they allow us to transform products and exponents into differences and linear forms, which are easier to manage algebraically. In the exercise, to solve \( e^{-0.21 a} = \frac{11.53}{14.7} \), we take the natural logarithm of both sides:\[ \ln(e^{-0.21 a}) = \ln\left(\frac{11.53}{14.7}\right) \]This step uses the property \( \ln(e^x) = x \), simplifying the left-hand side to:\[ -0.21 a = \ln\left(\frac{11.53}{14.7}\right) \]Now, our equation has been linearized, making it straightforward to solve for \( a \). The natural logarithm is a tool that enables us to "undo" the effect of an exponential function.

Unit Conversion

Unit conversion is a critical step when the answer's unit differs from the source data. In this problem, our calculation gives altitude in miles, but we need it to express it in feet.To convert miles into feet, remember that 1 mile is equal to 5280 feet. Thus, if you find an altitude of \( a \) miles, the conversion into feet is:\[ \text{Altitude in feet} = a \times 5280 \]This step ensures that you communicate the final result in terms of the units asked for in the problem statement. Moreover, converting units correctly is crucial in retaining the integrity of your calculated answer. After conversion, the altitude for Cheyenne, Wyoming, needs rounding to the nearest hundred feet, which leads to the answer being more practical and easier to interpret.

Problem-Solving Steps

The problem-solving process involves breaking down the problem into manageable steps, which makes it less daunting and more systematic. Here is a simplified breakdown:

• Understand the Problem: Identify what you are given and what you need to find. Here, you’re given the equation for atmospheric pressure and you need to find the altitude.
• Set Up the Equation: Substitute known values into the given equation to clarify what needs solving.
• Isolate the Exponential Term: Modify the equation to isolate the exponent on one side, which sets the stage for using logarithms.
• Solve for the Variable: Use natural logarithms to solve for the unknown variable.
• Convert Units: Ensure that your answer is in the desired unit by converting as necessary.
• Express Answer Appropriately: After calculations, adjust the final answer to match the required precision.

This structured approach not only leads to the correct answer but also helps demystify complex problems through logical progression.