Problem 98 - College Algebra with Modeling and Visualization

Question

Survival of Reindeer For all types of animals, the percentage that survive into the next year decreases. In one study, the survival rate of a sample of reindeer was modeled by \(S(t)=100(0.999993)^{t^{3}} .\) The function \(S\) outputs the percentage of reindeer that survive \(t\) years. (a) Bvaluate \(S(4)\) and \(S(15)\). Interpret the results. (b) Graph \(S\) in \([0,15,5]\) by \([0,110,10]\). Interpret the graph. Does the graph have a horizontal asymptote?

Step-by-Step Solution

Answer

(a) 99.96% survive after 4 years; 98.0% survive after 15 years. (b) The graph shows a decreasing survival rate and approaches a horizontal asymptote at 0.

Step 1: Evaluating S(4)

To find the survival rate after 4 years, we substitute \(t = 4\) into the survival function \(S(t)\): \[S(4) = 100(0.999993)^{4^3} = 100(0.999993)^{64}.\]Using a calculator, we find \[(0.999993)^{64} \approx 0.999552.\]Thus, \[S(4) = 100 \times 0.999552 = 99.9552.\]Therefore, about 99.96% of the reindeer survive after 4 years.

Step 2: Evaluating S(15)

Next, to find the survival rate after 15 years, substitute \(t = 15\) into \(S(t)\): \[S(15) = 100(0.999993)^{15^3} = 100(0.999993)^{3375}.\]Using a calculator, we calculate \[(0.999993)^{3375} \approx 0.980.\]Thus, \[S(15) = 100 \times 0.980 = 98.0.\]So, about 98.0% of the reindeer survive after 15 years.

Step 3: Graphing S(t)

To graph the function \(S(t)\), choose values of \(t\) from 0 to 15. Calculate \(S(t)\) for several points, for example at \(t = 0, 5, 10, 15\). Note that the function decreases over time. Plot these points: - \(t = 0\), \(S(0) = 100\)- \(t = 5\), \(S(5) \approx 99.93\)- \(t = 10\), \(S(10) \approx 99.80\)- \(t = 15\), \(S(15) = 98.0\).Connect the points smoothly, showing a decrease in survival rate.

Step 4: Analyzing the Graph

Observe from the graph that the function starts at 100% and decreases slightly with time, showing a survival rate decline. The graph tends towards a horizontal line as \(t\) increases. Yes, the graph has a horizontal asymptote at \(S(t) = 0\) because as time goes on, theoretically, the survival percentage will approach zero but never actually reach it.

Key Concepts

Survival Rate

The concept of survival rate is crucial in understanding how populations of animals or other entities persist over time. In the case of the reindeer, the survival rate is given by the function \( S(t)=100(0.999993)^{t^{3}} \). This function measures the percentage of a reindeer population that survives after a given number of years, \( t \).
The key takeaway is that as time goes by, not all individuals survive, and the percentage that does tends to decrease. In this particular study:

• At \( t = 4 \), it was found that approximately 99.96% survive, meaning a very high survival rate in the short term.
• At \( t = 15 \), approximately 98.0% survive, indicating a slight decline over the years.

This model provides a mathematical way to predict future survival rates based on past data, enabling researchers to make informed decisions about conservation efforts.

Graph Interpretation

Graph interpretation helps us visualize the survival pattern of reindeer over time. When you plot the survival function \( S(t) \) from \( t = 0 \) to \( t = 15 \), you see a curve that starts at 100% and gradually declines.
The graph reveals several important insights:

• The gradual decline illustrates the decrease in survival rate over the years.
• Key points like \( S(0) = 100 \), \( S(5) \approx 99.93 \), \( S(10) \approx 99.80 \), and \( S(15) = 98.0 \) mark specific time intervals.

This not only shows the percentage of reindeer surviving at given times but also indicates the continuous nature of decline. Graph interpretation is critical because visualizations make it easier to comprehend complex mathematical relationships.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches but never actually meets as time goes on. In the reindeer survival graph, as \( t \) approaches infinity, the function \( S(t) \) approaches a value closer to zero, but never quite reaches it.
This tells us:

• In the long run, survival approaches zero, indicating that nearly all reindeer eventually perish over an extended period.
• The concept of an asymptote helps model the idea that while survival rates decrease, they don’t become zero in a finite amount of time.

Understanding asymptotes is important because it allows for predictions about behavior over time, especially in growth-decay models.

Mathematical Modeling

Mathematical modeling refers to using mathematical formulas and functions to represent real-world phenomena. In this context, the function \( S(t)=100(0.999993)^{t^{3}} \) models the survival rate of reindeer over time.
Here are some key aspects:

• **Accuracy**: This exponential model closely approximates the observed data, allowing scientists to predict future trends effectively.
• **Predictability**: By inserting different values of \( t \), one can predict survival rates for different time periods without additional data collection.
• **Insights**: It helps understand factors affecting survival and assess interventions' potential impacts to improve populations.

Mathematical models are powerful because they simplify complex natural processes into manageable equations, aiding analysis and strategic planning.