Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

Match each equation with its model. (a) Exponential growth model (i) \(y=a e^{-b x}, b>0\) (b) Exponential decay model (ii) \(y=a+b \ln x\) (c) Logistic growth model (iii) \(y=\frac{a}{1+b e^{-r x}}\) (d) Gaussian model (iv) \(y=a e^{b x}, b>0\) (e) Natural logarithmic model (v) \(y=a+b \log _{10} x\) (f) Common logarithmic model (vi) \(y=a e^{-(x-b)^{2} / c}\)

Step-by-Step Solution

Answer

(a) matches with (iv), (b) matches with (i), (c) matches with (iii), (d) matches with (vi), (e) matches with (ii), (f) matches with (v).

Step 1: Match the Exponential Growth Model

Looking at the expressions, (iv) \(y=a e^{b x}, b>0\) can be classified as an Exponential growth model since the variable x is in the exponent and b>0 which ensures growth.

Step 2: Match the Exponential Decay Model

Expression (i) \(y=a e^{-b x}, b>0\) fits the Exponential decay model as it includes the variable x in the exponent with a negative sign, indicating decay.

Step 3: Identify the Logistic Growth Model

Expression (iii) \(y=\frac{a}{1+b e^{-r x}}\) fits the Logistic growth model which often includes an equation in the form of a fraction where an exponential expression is in the denominator.

Step 4: Recognize the Gaussian Model

Expression (vi) \(y=a e^{-(x-b)^{2} / c}\) is classified as a Gaussian model. It involves the exponential of a squared expression, a common characteristic of Gaussian models.

Step 5: Pinpoint the Natural Logarithmic Model

Expression (ii) \(y=a+b \ln x\) matches the Natural logarithmic model since it involves a natural logarithm (logarithm with base e).

Step 6: Find the Common Logarithmic Model

Expression (v) \(y=a+b \log _{10} x\) fits the Common logarithmic model as it includes a logarithm with base 10.

Key Concepts

Exponential Decay Model

In the context of mathematical models, the exponential decay model is used to describe processes that decrease over time. These processes might include radioactive decay, cooling of hot liquids, or depreciation of assets.
The general form of the exponential decay equation is \( y = a e^{-bx} \), where:

• \(a\) represents the initial value.
• \(b\), a positive constant, determines the rate of decay.
• \(x\) is the independent variable, often representing time.

The negative exponent \(-bx\) is key in exponential decay, as it ensures the value of \(y\) decreases as \(x\) increases.
This model is widely applicable in various scientific fields, providing a precise mathematical representation of how quantities diminish.

Logistic Growth Model

The logistic growth model is used to describe growth that is initially exponential, but slows down as the process reaches a maximum capacity. This model is particularly useful in biological contexts, such as population growth where resources are limited.
The standard form of the logistic growth equation is \( y = \frac{a}{1 + be^{-rx}} \), where:

• \(a\) is the carrying capacity, or the maximum sustainable value.
• \(b\) is a constant that affects the curve's initial increase.
• \(-rx\) in the exponent ensures the decreasing rate of growth over time.

As \(x\) increases, the value of \(y\) approaches \(a\), reflecting the saturation point. This model highlights the realistic constraints on growth arising from limited resources.

Gaussian Model

The Gaussian model, also known as the normal distribution, is represented by an equation that generates a symmetric bell-shaped curve about the mean value. It is widely used in statistics to represent real-valued random variables with unknown distributions.
The Gaussian model has the form \( y = a e^{-(x-b)^{2}/c} \), where:

• \(a\) controls the height of the peak.
• \(b\) is the mean or the center of the distribution.
• \(c\) affects the width of the bell curve.

The exponential of a squared expression \(-(x-b)^{2}\) offers the distinctive symmetry of the Gaussian curve. This model is crucial in fields requiring accurate predictions and analysis, such as finance, physics, and health sciences.

Natural Logarithmic Model

The natural logarithmic model uses the natural logarithm, which has a base \(e\). This model is especially useful in contexts involving continuous growth processes, such as compound interest or biological growth.
The model typically takes the form \( y = a + b \ln x \), where:

• \(a\) sets the baseline or starting point.
• \(b\), a coefficient, influences the steepness of the curve.
• \( \ln x\) indicates the natural logarithm of \(x\).

Since the natural logarithm grows at a diminishing rate as \(x\) increases, this model aptly describes processes that speed up initially and then level off. Understanding this model aids in mastering various natural patterns of change.

Common Logarithmic Model

The common logarithmic model uses the base 10 logarithm, which is common in measurement systems and logarithm tables. It often helps in solving problems where the logarithm of a quantity varies linearly with another variable.
The model is expressed as \( y = a + b \log_{10} x \), where:

• \(a\) is a constant that sets the initial value.
• \(b\) determines how quickly the logarithmic value increases.
• \( \log_{10} x \) represents the base 10 logarithm of \(x\).

This model is particularly suited for data spanning several orders of magnitude, simplifying complex calculations. It is prominent in scientific fields such as acoustics, astronomy, and geology.