Problem 52 - Precalculus

Question

Consider the quotient \(Q=\frac{P_{1} e^{r_{1} t}}{P_{2} e^{r_{2} t}}\) of two models of exponential growth. a. If \(r_{1}>r_{2},\) what can you say about \(Q ?\) b. If \(r_{1}

Step-by-Step Solution

Answer

a. Q grows exponentially if \(r_1 > r_2\). b. Q decays exponentially if \(r_1 < r_2\).

Step 1: Analyze the Expression for Q

The expression for the quotient is given as \[ Q = \frac{P_{1} e^{r_{1} t}}{P_{2} e^{r_{2} t}}. \]This can be rewritten into a simpler form by factoring out the exponential components:\[ Q = \left(\frac{P_{1}}{P_{2}}\right) \times e^{(r_{1} - r_{2})t}. \]

Step 2: Evaluate the Case When r1 > r2

If \(r_{1} > r_{2}\), then \(r_{1} - r_{2} > 0\). Therefore, the expression:\[ e^{(r_{1} - r_{2})t} \]represents exponential growth since the exponent is positive.As \(t\) increases, \(Q\) will grow larger because the exponential growth term dominates the expression.

Step 3: Evaluate the Case When r1 < r2

If \(r_{1} < r_{2}\), then \(r_{1} - r_{2} < 0\). Therefore, the expression:\[ e^{(r_{1} - r_{2})t} \]represents exponential decay since the exponent is negative.As \(t\) increases, \(Q\) will decrease because the exponential decay term will cause the quotient to shrink.

Key Concepts

Quotient of Exponentials

When dealing with exponential functions, we may often encounter scenarios where we need to compare or manipulate exponential expressions. One such example is when we form a quotient of exponential functions. Consider the expression. \[ Q = \frac{P_{1} e^{r_{1} t}}{P_{2} e^{r_{2} t}} \] This form is not just a simple division of numbers, but it contains an exponential component. By breaking this expression into simpler parts, we can rewrite it as: \[ Q = \left(\frac{P_{1}}{P_{2}}\right) \times e^{(r_{1} - r_{2})t} \]Here, we factored the exponential parts to clearly identify the role of the exponents. This step simplifies the understanding of how changes in time \(t\) affect the behavior of \(Q\). Whether \(Q\) increases or decreases depends on whether \(r_1 > r_2\) or \(r_1 < r_2\). Understanding this is crucial for comparing and analyzing exponential growth or decay scenarios.

Exponential Models

Exponential models are mathematical expressions that describe exponential growth or decay over time. These models are characterized by a constant percentage growth rate or decay rate applied to a base quantity, usually represented as: \[ P(t) = P_0 e^{rt} \]Where \(P(t)\) is the quantity at time \(t\), \(P_0\) is the initial quantity, and \(r\) is the rate of growth or decay. Exponential growth occurs when the growth rate \(r\) is positive, resulting in a model that increases as time progresses. On the other hand, exponential decay takes place when \(r\) is negative, causing the modeled quantity to decrease over time. In the context of our exercise, analyzing two models given by \(P_1 e^{r_1 t}\) and \(P_2 e^{r_2 t}\) help us understand how different growth rates \(r_1\) and \(r_2\) influence comparisons between quantities described by these models.

Comparing Growth Rates

Comparing the growth rates of different exponential models provides insight into how each model behaves over time. Suppose we have two models with exponents \(r_1\) and \(r_2\) in our expression for \(Q\): \[ Q = \left(\frac{P_{1}}{P_{2}}\right) \times e^{(r_{1} - r_{2})t} \]If \(r_1 > r_2\), the term \(e^{(r_1 - r_2)t}\) ensures that \(Q\) experiences exponential growth. This is because the difference \((r_1 - r_2)\) in the exponent is positive, leading \(Q\) to increase as time \(t\) advances. Conversely, if \(r_1 < r_2\), the quotient transitions into exponential decay since the exponential term \((r_1 - r_2)\) becomes negative, causing \(Q\) to shrink over time. By comparing these rates, you can predict and quantify the behavior of the models, making it valuable for real-world applications where growth trends or declines are essential.