Problem 39 - Algebra 2

Question

Blotocy Suppose an amoeba divides into two amoebas once every hour. How long would it take for a single amoeba to become a colony of 4096 amoebas?

Step-by-Step Solution

Answer

It takes 12 hours for a single amoeba to become a colony of 4096 amoebas.

Step 1: Understand the Growth Pattern

The amoeba divides into two every hour. This means the number of amoebas doubles every hour.

Step 2: Represent the Growth Mathematically

If we start with 1 amoeba, the number of amoebas after \( n \) hours is given by the exponential equation \( 2^n \). We need to find \( n \) such that \( 2^n = 4096 \).

Step 3: Equate and Solve the Exponential Equation

Set up the equation \( 2^n = 4096 \). We need to find the power of 2 that equals 4096.

Step 4: Use Logarithms to Solve for n

To solve for \( n \), take the logarithm base 2 of both sides: \( n = \log_2(4096) \).

Step 5: Calculate \( \log_2(4096) \)

Evaluating \( \log_2(4096) \) means finding which power of 2 gives 4096. Since \( 4096 = 2^{12} \), it follows that \( n = 12 \).

Key Concepts

Doubling Time

Doubling time is the amount of time it takes for a quantity to double in size or value. In the context of growth, such as our amoeba example, doubling time is crucial to understand how quickly the population expands. If we start with one amoeba, and it splits into two every hour, the doubling time is just one hour. This type of growth is characteristic of exponential growth, where the rate of increase is proportional to the current amount. In mathematical terms, exponential growth can be described with the formula:

• Initial Amount: Let's denote the initial number of amoebas as 1.
• Doubling every hour: This means in every hour, the number of amoebas is multiplied by 2.

So, after one hour, you have 2 amoebas; after two hours, you have 4 amoebas, and so on. The simplicity of this growth model makes it a key concept in understanding how populations and investments grow over time.

Exponential Equation

An exponential equation is one where variables appear as exponents. In our situation, the number of amoebas forms an exponential equation because the number doubles every hour. This type of equation follows the general form:

• Exponential Equation: \( a^n = y \)
• a is the base, which is 2 (since the amoebas double).
• n represents the time elapsed.
• y is the number of amoebas after n hours.

In our example, we use the exponential equation \( 2^n = 4096 \). We must find which power of 2 results in 4096. Exponential equations such as these are typically solved using logarithms because logarithms can "undo" exponents and reveal the unknown variable.

Logarithms

Logarithms are mathematical tools that help us solve equations where the unknown is an exponent. When dealing with exponential growth, we often need to find the power or time involved. In our scenario, we solved the equation \( 2^n = 4096 \) using a logarithm.Logarithm basics involve:

• Definition: If \( a^n = y \), then \( n = \log_a(y) \)
• Base: For our situation, we use base 2 (because of doubling).

Taking the logarithm of both sides of \( 2^n = 4096 \) helps us find that \( n = \log_2(4096) \). By evaluating this, we find similarities with powers of 2. Through calculation, we determine \( 4096 = 2^{12} \). Thus, \( n = 12 \) hours. Logarithms simplify exponential growth calculations, enabling us to solve problems practically. They are especially helpful in cases of rapid growth, like those involving our doubling amoebas.