Problem 10 - Algebra for College Students

Question

A fungus culture growing under controlled conditions doubles in size each day. How many units will the culture contain after 7 days if it originally contains 4 units? 512

Step-by-Step Solution

Answer

After 7 days, the culture will contain 512 units.

Step 1: Understand the Given Information

The fungus culture starts with an initial amount of 4 units and doubles every day. We need to calculate how many units there will be after 7 days.

Step 2: Identify the Doubling Process

On each day, the amount of the fungus is doubled. Mathematically, this can be expressed as the initial amount multiplied by 2 to the power of the number of days.

Step 3: Set Up the Equation

The equation is given by:\[N = 4 \times 2^d\]where \(N\) is the number of units on the \(d\)th day, and \(d\) is the number of days which is 7 in this problem.

Step 4: Calculate the Number of Units

Substitute \(d=7\) into the equation:\[N = 4 \times 2^7\]Calculate \(2^7 = 128\), and then multiply by 4:\[N = 4 \times 128 = 512\]

Step 5: Conclusion

After performing the calculations, we find that after 7 days, the fungus culture will contain 512 units.

Key Concepts

Doubling Time

Doubling time is a crucial concept when we talk about exponential growth. It's the period needed for a quantity to double in size or amount. When a population or phenomenon grows exponentially, it adds the same percentage of its current size at every step. This results in rapid increases over time.
For instance, in the context of the given problem, the fungus culture doubles each day. This daily doubling can be described by saying the doubling time is 1 day. Understanding this simple timeframe helps us predict how the culture size changes over several days.

• Doubling time is consistent for any exponentially growing quantity.
• It provides a way to anticipate future growth without extensive calculations.
• It's always expressed in the same time units as the growth process (e.g., days, months).

When faced with similar problems, identifying the doubling time can simplify your approach and give you a clearer perspective on how quickly things are scaling.

Exponential Function

The exponential function is a mathematical expression crucial for modeling situations where quantities multiply at consistent rates over equal intervals. In essence, it helps describe growth patterns like the one in our example with the fungus culture.
An exponential function generally takes the form of:\[ y = a \times b^x \]- Here, \(a\) is the initial amount.- \(b\) is the base or growth factor.- \(x\) is the exponent, representing the number of time intervals.
In our exercise, the exponential function used is:\[ N = 4 \times 2^d \]- The initial amount \(a\) is 4 units.- The growth factor \(b\) is 2, indicating doubling.- \(d\) is the time in days. This reflects how quantities evolve rapidly under exponential growth, providing a formulaic approach to predict changes over time.
Exponential functions can model various real-world phenomena, from population growth to radioactive decay, highlighting their versatile nature in problem-solving.

Problem Solving in Algebra

Problem-solving in algebra often involves translating a word problem into a mathematical equation and solving it systematically. Here's how we can break down the procedure using our exercise as a guide.
First, understand the problem at hand. Here, acknowledging that the fungus doubles daily gives us the insight needed to set up an appropriate model or equation. Next, represent known values and relationships with symbols, which might involve variables and operations like multiplication or exponentiation.

• Define the variables: In our case, \(d\) represented the number of days.
• Formulate the equation: Using known values, the equation was created as \(N = 4 \times 2^d\).
• Solve the equation: Substitute the specific scenario details. For 7 days, \(d=7\), which leads to \(N = 4 \times 128 = 512\).

Having a step-by-step approach helps demystify seemingly complex problems, ensuring clarity and precision in algebraic problem-solving. Always take the time to interpret your results in the context of the problem, confirming they align with expectations.