Problem 353 - Calculus Volume 1

Question

True or False? If true, prove it. If false, find the true answer. If bacteria increase by a factor of 10 in 10 hours, how many hours does it take to increase by 100\(?\)

Step-by-Step Solution

Answer

False. It takes 20 hours to increase by a factor of 100.

Step 1: Understand Growth Pattern

Since bacteria grow by a factor of 10 in 10 hours, this indicates exponential growth. This means the population multiplies by 10 every 10 hours.

Step 2: Establish Growth Function

If we let the initial bacteria count be \( P_0 \), the population after \( t \) hours can be described by the equation \( P(t) = P_0 imes 10^{t/10} \).

Step 3: Formulate Equation for 100-fold Increase

To find how many hours it takes to increase by 100 times, set \( P(t) = 100 \times P_0 \). Substitute into our growth function: \( 100 \times P_0 = P_0 imes 10^{t/10} \).

Step 4: Solve for Time \( t \)

Divide both sides by \( P_0 \): \[ 100 = 10^{t/10} \]Take the logarithm of both sides: \[ \log_{10}(100) = \log_{10}(10^{t/10}) \]Simplify to find \( t \): \[ 2 = \frac{t}{10} \]\[ t = 20 \]Thus, it takes 20 hours to increase by 100 times.

Key Concepts

Bacterial Growth

Bacterial growth is a fascinating example of exponential growth in biology. When we talk about bacterial growth under ideal conditions, we often describe how bacteria double at regular intervals. This means if you start with a certain amount of bacteria, after a set period of time, that amount will have multiplied. In the exercise provided, bacteria increase by a factor of 10 in 10 hours. This tells us that every 10 hours, the bacterial population is 10 times what it was at the start of that duration. This exponential increase is typical in bacterial growth under optimal conditions where resources aren't a limiting factor. The growth pattern can be observed in a specific form of exponential function, helping us model and predict how a bacterial population will behave over time. Understanding and predicting this growth helps in various fields, such as medicine and food science, where controlling bacterial growth is crucial.

Logarithms

Logarithms are a powerful mathematical tool used to solve exponential equations, like those encountered in bacterial growth problems. A logarithm essentially tells us the power or exponent that a base number must be raised to, in order to produce a given number. For example, in the bacterial growth problem, we used base 10 logarithms to solve for time (t) when the population increases by 100 times:

• The equation to solve was: \( 100 = 10^{t/10} \)
• By taking the logarithm of both sides, we found: \( \log_{10}(100) = \log_{10}(10^{t/10}) \)
• The property of logarithms being used here simplifies to: \( 2 = \frac{t}{10} \)

This makes logarithms incredibly useful for dealing with equations where the variable is an exponent, particularly in exponential growth scenarios.

Exponential Functions

Exponential functions are mathematical functions in the form of \( f(x) = a \times b^x \), where \( a \) is a constant, \( b \) is the base of the exponential, and \( x \) is the exponent. They model situations where growth or decay happens at a consistent percentage rate over time. In our exercise, the exponential function was used to model the bacterial growth over time. The function given was:\[ P(t) = P_0 \times 10^{t/10} \]Here, \( P_0 \) represents the initial bacteria count, \( 10 \) is the growth factor over a 10-hour span, and \( t/10 \) shows how time affects the growth period. Such functions are crucial in understanding how variables interact over time, especially when dealing with rapid growth scenarios like populations, investments, or any situation where change happens exponentially. By understanding exponential functions, you can predict and analyze the underlying patterns of various real-world processes.