Problem 58 - Calculus for Biology and Medicine

Question

Assume that a population size at time \(t\) is \(N(t)\) and that \(N(t)=20 \cdot 2^{t}, t \geq 0\). (a) Find the population size at time \(t=0\). (b) Show that \(N(t)=20 e^{t \ln 2}, t \geq 0\) (c) How long will it take until the population size reaches \(1000 ?\) \([\) Hint \(:\) Find \(t\) so that \(N(t)=1000 .]\)

Step-by-Step Solution

Answer

(a) 20. (b) Shown. (c) 5.644 units of time.

Step 1: Calculating Population at t=0

At time \(t=0\), the formula given is \(N(t) = 20 \cdot 2^t\). Replace \(t\) with 0 to find \(N(0)\): \[ N(0) = 20 \cdot 2^0 = 20 \cdot 1 = 20. \] Thus, the population size at time \(t=0\) is 20.

Step 2: Converting the Expression to Exponential Form

To demonstrate that \(N(t) = 20 e^{t \ln 2}\), start with the original expression: \[ N(t) = 20 \cdot 2^t. \] Using the property \(a^b = e^{b \ln a}\), rewrite \(2^t\) as \(e^{t \ln 2}\): \[ 2^t = e^{t \ln 2}. \] Substitute back into the equation for \(N(t)\): \[ N(t) = 20 \cdot e^{t \ln 2}. \] Therefore, it is confirmed that \(N(t) = 20 e^{t \ln 2}\).

Step 3: Finding Time When Population Reaches 1000

To find the time \(t\) when \(N(t) = 1000\), set up the equation: \[ 20 \cdot 2^t = 1000. \] Divide both sides by 20: \[ 2^t = 50. \] Take the logarithm base 2 of both sides: \[ t = \log_2{50}. \] Using the change of base formula, \(\log_2{50} = \frac{\ln{50}}{\ln{2}}\). Calculate using a calculator: \[ t \approx \frac{\ln{50}}{\ln{2}} \approx 5.644. \] Thus, it will take approximately 5.644 time units for the population to reach 1000.

Key Concepts

Exponential Functions

Exponential functions are crucial in modeling situations where growth or decay happens at a constant rate. In our population growth problem, the function \(N(t) = 20 \cdot 2^t\) indicates an exponential growth model. Let's break it down further.

1. **Understanding the Formula**
- The function starts with a constant multiplier, 20, known as the initial population size at time \(t = 0\). - The term \(2^t\) represents growth factor with base 2, implying the population doubles for every unit increase in time \(t\).

2. **Characteristics of Exponential Growth**
- Exponential growth means as time progresses, the population increases exponentially rather than linearly. - This kind of growth is significant in biological sciences, where bacteria populations, for instance, grow exponentially.

Exponential functions provide a powerful way to predict future behavior for processes that grow or decay at constant percentage rates. Recognizing these patterns helps in understanding real-world phenomena.

Logarithms

Logarithms are the mathematical inverse of exponentiation and are key to solving equations involving exponential functions, like the one in our exercise.

1. **Understanding Logarithms**
- Logarithms solve for the exponent in an equation of the form \(a^x = b\). - If you have \(2^t = 50\), applying a logarithm helps isolate and solve for \(t\).

2. **Base Change and Applications**
- In the exercise, we used the base-2 logarithm to solve \(t = \log_2{50}\), but calculators commonly offer natural logarithms, so we employ the change of base formula: \[\log_2{50} = \frac{\ln{50}}{\ln{2}}\].
- Mastering logarithms makes it easier to tackle exponential equations, routinely used in fields like biology for determining time intervals of growth phases or chemical reaction rates.

Logarithms help unlock complexities in growth models, making them indispensable in quantitative predictions and scientific calculations.

Calculus for Life Sciences

Calculus is a fundamental tool in life sciences for modeling and analyzing continuous change, such as population growth over time. In this exercise, understanding calculus principles enhances our analysis of exponential growth.

1. **Differential Calculus**
- Differential calculus focuses on the concept of derivatives, which represent how a function changes at any given point. - In the context of population growth, we'd use derivatives to calculate the rate of change in population over time.

2. **Integral Calculus and Applications**
- Integral calculus, on the other hand, involves determining the cumulative total or area under a growth curve over time. - By integrating the population function, scientists estimate the total population accumulated over specific time periods.

In life sciences, calculus allows for a deeper understanding and more precise predictions of biological phenomena. It plays a critical role in interpreting and managing dynamic changes in data, crucial for research and application in areas such as ecology, pharmacology, and population dynamics.