Problem 12 - College Algebra

Question

Rewrite the exponential model \(A(t)=1550(1.085)^{x}\) as an equivalent model with base \(e\). Express the exponent to four signifi ant digits.

Step-by-Step Solution

Answer

The equivalent model is \(A(x) = 1550 \, e^{0.08121x}\).

Step 1: Identify the General Form

The given exponential model is \(A(x) = 1550(1.085)^x\). The general form for an exponential function with base \(a\) is \(A(x) = C \, a^x\). We need to rewrite it in terms of the natural base \(e\), which has the form \(A(x) = C \, e^{kx}\), where \(k\) is a constant we need to find.

Step 2: Set Equivalent Expressions

To rewrite the model using base \(e\), set \((1.085)^x = e^{kx}\). This implies that \(1.085 = e^k\). We need to find \(k\) so that both expressions are equivalent.

Step 3: Solve for k

Take the natural logarithm on both sides of the equation \(1.085 = e^k\) to solve for \(k\): \[\ln(1.085) = \ln(e^k)\]Since \(\ln(e^k) = k\), it follows that:\[k = \ln(1.085)\]

Step 4: Calculate k

Calculate \(k\) using your calculator:\[k = \ln(1.085) \approx 0.081214\]Make sure to round \(k\) to four significant digits: \(k \approx 0.08121\).

Step 5: Rewrite the Model

Now replace the original model, using the base \(e\):\[A(x) = 1550 \, e^{0.08121x}\]This is the equivalent model with base \(e\).

Key Concepts

Exponential Model

An exponential model is a mathematical representation of situations where quantities change exponentially over time. In an exponential model such as \( A(t) = 1550(1.085)^x \), the expression describes a situation where the amount \( A(t) \) grows by a constant multiplier, 1.085, for each unit increase in \( x \). This base, in this case, 1.085, tells us the growth factor of the model. Exponential models are widely used in various fields like finance for compound interest calculations, biology for population growth models, and physics for radioactive decay.

When working with exponential models, it's crucial to understand the behavior depicted by the base value. If the base is greater than 1, the model represents growth. If it's between 0 and 1, it signifies decay.

• Base greater than 1: Exponential Growth
• Base between 0 and 1: Exponential Decay

Understanding exponential models unlocks insights into patterns of change over time, making it an important concept in mathematics and real-world applications.

Base Change

Base change in exponential functions involves converting the model into a different base to simplify calculations or analysis. In the given exercise, we change the base from 1.085 to the natural base \( e \). This process makes mathematical manipulation easier, especially in calculus, allowing for more straightforward differentiation and integration.

To achieve this, the principle of equivalence is applied: \( (a)^x = e^{kx} \). Consequently, by setting \( a = e^k \), we find \( k \) by taking the natural logarithm: \( k = \ln(a) \).

• Specify the original base \( a \).
• Set \( a^x = e^{kx} \).
• Solve \( k = \ln(a) \).

Changing the base is a powerful technique that is indispensable when working with exponential functions, particularly when converting to the natural base \( e \), as it simplifies many processes in higher mathematics.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. In the context of the exercise, the natural logarithm is used to find the value of \( k \) when rewriting an exponential model to have base \( e \). This is done by applying the property of logarithms that allows us to solve equations in the form \( e^k = a \).

The natural logarithm has unique properties that make it especially useful in calculus and differential equations, such as the fact that the derivative of \( \ln(x) \) is \( 1/x \). This makes solving equations and modeling growth or decay simple and straightforward.

• \( \ln(e) = 1 \)
• \( \ln(1) = 0 \)
• Derivative: \( \frac{d}{dx} \ln(x) = \frac{1}{x} \)

Using the natural logarithm in rewriting models or solving exponential equations is an essential skill, making complex problems easier to manage and solve.

Significant Digits

Significant digits, also known as significant figures, are used to express the precision of a number. They are critical in preserving the accuracy of scientific and mathematical calculations. In the context of this exercise, the value of \( k \) was rounded to four significant digits, which is a standardized practice when expressing results in scientific and mathematical contexts.

Significant digits include all non-zero numbers, any zeros between significant figures, and any trailing zeros in the decimal portion. Understanding and using significant digits correctly ensures clarity and consistency in numerical information.

• Leading zeros are not significant.
• Trailing zeros in a number with a decimal are significant.
• When performing calculations, the number of significant digits in the final answer should reflect the least precise measurement.

Providing results with the correct number of significant digits is crucial as it conveys the precision of the measurements and calculations. This ensures that any estimations or subsequent calculations remain as accurate as the original data allows.