Problem 9 - Calculus for Biology and Medicine

Question

In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. Suppose \(N_{t}=100 \cdot 2^{t}, t=0,1,2, \ldots\), and one unit of time corresponds to 2 hours. Determine the amount of time it takes the population to triple in size.

Step-by-Step Solution

Answer

It takes approximately 3.17 hours for the population to triple.

Step 1: Understand the Problem

We are given the formula for a population growth over time: \( N(t) = 100 \cdot 2^{t} \). Our goal is to determine how long it takes for this population to triple in size.

Step 2: Set Up the Equation for Tripled Population

Since the initial population is \( N(0)=100 \), tripling the population means we want \( N(t) = 300 \). Use the formula: \( N(t) = 100 \cdot 2^{t} = 300 \).

Step 3: Simplify the Equation

Divide both sides of the equation \( 100 \cdot 2^{t} = 300 \) by 100 to isolate \( 2^{t} \). This gives us: \[ 2^{t} = 3 \].

Step 4: Solve for \( t \) Using Logarithms

Take the logarithm of both sides of the equation \( 2^{t} = 3 \). Use base 2 logarithm for simplicity: \[ \log_{2}(2^{t}) = \log_{2}(3) \]. Thus \( t = \frac{\log_{10}(3)}{\log_{10}(2)} \) which approximately equals 1.585.

Step 5: Convert \( t \) to Time in Hours

Since \( t \) is in units where one unit equals 2 hours, multiply \( t \) by 2 to convert to hours: \[ 1.585 \times 2 = 3.17 \text{ hours} \].

Key Concepts

Exponential Growth

Exponential growth is a powerful concept where something increases at a rate proportional to its current value. This is common in biological populations, investments, and even some physical processes. In exponential growth, the quantity grows by a constant factor over equal time intervals.

A basic formula for exponential growth is:

• \[ N(t) = N_0 \cdot b^{t} \]

where:

• \( N(t) \) is the quantity at time \( t \)
• \( N_0 \) is the initial quantity
• \( b \) is the base of the exponential, which represents the growth factor
• \( t \) is the time elapsed

In our specific example, the growth factor is 2, meaning the population doubles every time period \( t \). This leads to rapid growth, which can sometimes outpace other processes like resource availability or space.

Logarithms

Logarithms are the inverses of exponential functions, helping us solve equations where the variable is an exponent. In simpler terms, if you know the result of an exponential equation, logarithms help find the time it took for the growth to reach that size.

For equation solving, we shift from an exponential form to a logarithmic form:

• If \( b^x = y \), then \( x = \log_b(y) \)

In the given problem, we used logarithms to solve \( 2^{t} = 3 \). This can be expressed with base-10 logarithms:

• \[ t = \frac{\log_{10}(3)}{\log_{10}(2)} \]

Understanding logs is crucial for converting between exponential growth rates and finding the actual time \( t \) needed for certain growth benchmarks, like tripling a population.

Time Calculation

Once we've determined \( t \) in the growth equation, understanding the actual time in real-world units is crucial. In this problem, a single unit of \( t \) corresponds to 2 hours. Once we've calculated \( t \) as 1.585, we need to convert this into hours.

To transform \( t \) to a real-world timing, multiply \( t \) by the time per unit:

• \[ 1.585 \times 2 = 3.17 \text{ hours} \]

Thus, it takes approximately 3.17 hours for the population to triple. This conversion ensures we can apply the mathematical solution to actual scenarios, assisting in planning or further analysis of the growth scenario.