Problem 12 - Calculus for Biology and Medicine

Question

Suppose that the biomass of a population at time \(t\) is given by $$ B(t)=\frac{32.00 t}{17.00+t}, \quad t \geq 0 $$ (a) Use a calculator to confirm that \(B(10)\) is approximately 1.185185. Considering the function \(B(t)\), how many significant figures should you report in your answer? (b) Discuss the use of continuous functions in this problem.

Step-by-Step Solution

Answer

(a) For four significant figures, report 11.85. (b) Continuous functions model smooth population dynamics, suitable for predictions.

Step 1: Plug in the Value

For part (a), we need to calculate the biomass at time \( t = 10 \). Substitute \( t = 10 \) into the function \( B(t) \): \[ B(10) = \frac{32.00 \times 10}{17.00 + 10} \] Calculating this will help us verify the approximation.

Step 2: Perform the Calculation

Compute the expression from Step 1:\[ B(10) = \frac{32.00 \times 10}{27.00} = \frac{320.00}{27.00} \approx 11.85185 \] Thus, \( B(10) \) is indeed approximately 11.85185.

Step 3: Consider Significant Figures

Convert the calculated result to match significant figures appropriately. Here, both 32.00 and 27.00 have four significant figures, which suggests reporting the answer with four significant figures: 11.85.

Step 4: Discuss Continuous Functions

For part (b), continuous functions like \( B(t) \) help model populations over time smoothly. They allow for predictions of biomass at any time \( t \geq 0 \) and can handle dynamic changes in population efficiently by providing a continuous output for any input \( t \). This smooth, continuous model is crucial for understanding real-world scenarios where discrete changes are impractical.

Key Concepts

Significant Figures

Significant figures are the digits in a number that convey meaningful information regarding its precision. In mathematical calculations, they help communicate the precision of measurements or values used, ensuring uniformity in scientific communication. When dealing with the function \( B(t) = \frac{32.00 t}{17.00 + t} \), the numbers used in calculations are given with specific precision, noted by their significant figures.In this exercise, the numbers 32.00 and 17.00 both contain four significant figures. This choice of four significant figures indicates a level of precision aimed at maintaining consistency in the calculations across the board. It is important to ensure that the result matches the precision given in the problem. Hence, when you compute the value of \( B(10) \), ending up with 11.85185, using four significant figures is appropriate: 11.85.Properly using significant figures prevents the impression of unwarranted precision. It aligns the reported answer's precision with that of the inputs used, which is crucial when communicating scientific findings or results.

Continuous Functions

Continuous functions are mathematical tools used to describe how quantities change smoothly without abrupt breaks or jumps. Unlike functions that output discrete points, continuous functions provide a value for every single point on their input range. This is especially useful in contexts like this exercise, where we model biomass over time using \( B(t) = \frac{32.00 t}{17.00 + t} \).Using a continuous function allows us to predict the biomass for any time value, \( t \geq 0 \), providing insights into how the population changes dynamically over time. This means we can understand smooth transitions and gradual changes of population size. Such models are helpful, as in real-world biology, changes in population aren't usually discrete; they occur in a fluid transition.Continuous functions are vital for precision modeling in fields such as ecology and environmental science, where they help in predicting trends and formulating policies based on ongoing and future changes.

Population Dynamics

Population dynamics is a branch of life sciences that explores the changes in population sizes and compositions over time. This concept is pivotal in ecological studies where understanding how populations grow, shrink, and interact is crucial.In this exercise's context, the given function describes how biomass changes with time, forming a foundation for modeling population dynamics mathematically. The function \( B(t) = \frac{32.00 t}{17.00 + t} \) provides a formulaic representation of how biomass, which is an indicator of population size, evolves.Understanding population dynamics helps in managing wildlife resources, predicting environmental impacts, and devising conservation strategies. Models based on continuous functions, like the one here, allow researchers to simulate various scenarios, helping make predictions about the future of particular species under different conditions.Clear grasp of such dynamics is essential for effective biodiversity management and understanding the ecosystem's carrying capacity, ensuring both scientific advancement and real-world applicability in dealing with ecological challenges.