Problem 90 - College Algebra with Modeling and Visualization

Question

Bacteria Growth their size and divide every 40 minutes. (a) Write a recursive sequence that describes this growth where each value of \(n\) represents a 40 -minute interval. Let \(a_{1}=300\) represent the initial number of bacteria per milliter. Find the first five terms. (b) Determine the number of bacteria per milliliter after 10 hours have elapsed. (c) Is this sequence arithmetic or geometric? Explain.

Step-by-Step Solution

Answer

(a) 300, 600, 1200, 2400, 4800. (b) 9,830,400 bacteria per milliliter. (c) The sequence is geometric.

Step 1: Define the Recursive Sequence

To express the growth of bacteria, we observe that each 40-minute interval results in a doubling of the bacteria count. Therefore, the recursive sequence formula will be \( a_n = 2 \cdot a_{n-1} \). This means each term is twice the previous term.

Step 2: Establish Initial Conditions

We know that \( a_1 = 300 \), as given by the problem. This represents the number of bacteria initially present.

Step 3: Calculate the First Five Terms

Using the recursive formula \( a_n = 2 \cdot a_{n-1} \) and the initial condition, we find:1. \( a_1 = 300 \)2. \( a_2 = 2 \cdot 300 = 600 \)3. \( a_3 = 2 \cdot 600 = 1200 \)4. \( a_4 = 2 \cdot 1200 = 2400 \)5. \( a_5 = 2 \cdot 2400 = 4800 \)

Step 4: Find Number of Bacteria After 10 Hours

10 hours is equivalent to 600 minutes. Since each interval is 40 minutes, there are \( \frac{600}{40} = 15 \) intervals in 10 hours.Using our sequence, we need to find \( a_{16} \), as \( a_1 \) is already the first interval: \( a_{16} = 2^{15} \cdot a_1 = 2^{15} \cdot 300 \). We calculate: \( 2^{15} = 32768 \) \( a_{16} = 32768 \cdot 300 = 9830400 \).

Step 5: Identify Sequence Type

This sequence is geometric because each term is the previous term multiplied by a fixed number, specifically multiplied by 2 in this case.

Key Concepts

Bacteria Growth

Bacteria grow through a process of division where a single bacterium splits into two. This process can be quite rapid, leading to exponential growth under ideal conditions. In the context of our exercise, the bacteria double every 40 minutes. This means:

• Each interval marks the population doubling.
• If you start with 300 bacteria, after 40 minutes, you'll have 600.
• This doubling is consistent for every subsequent interval.

This rapid growth can be influenced by various factors such as temperature, nutrient availability, and other environmental conditions. In controlled environments, like lab settings, this kind of growth is predictable, forming an ideal scenario for mathematical modeling.

Geometric Sequence

A geometric sequence is a series of numbers with a constant ratio between consecutive terms. In simple words, each term after the first is found by multiplying the previous one by a fixed, non-zero number. For the bacterial growth:

• The doubling pattern forms a geometric sequence.
• Here, the common ratio or factor is 2, reflecting the population doubling every 40 minutes.

This consistent multiplicative step differentiates geometric sequences from arithmetic ones, where the addition of a fixed number separates terms. Understanding this property is crucial, as it allows for the prediction of long-term growth patterns using simple calculations.

Initial Conditions

In mathematics, initial conditions act as the starting point for sequences or problems. They are the foundation upon which subsequent calculations are built. In our problem:

• The initial condition is the number of bacteria at the start, given as 300.
• This serves as the base case for developing the recursive formula.

Initial conditions are essential because, without them, sequences cannot progress; they provide the first piece of the puzzle, which dictates how future pieces (terms) will be shaped. The accuracy of calculations, particularly in recursive sequences, heavily relies on these initial conditions.

Term Calculation

Term calculation in a sequence involves determining the value of specific terms based on a given rule or formula. In our case, the rule is a recursive formula. This formula is:\[ a_n = 2 \cdot a_{n-1} \]With the starting value \( a_1 = 300 \), we apply the recursive rule to calculate successive terms:

• To find the second term \( a_2 \), multiply the first term by 2, resulting in 600.
• Continue this pattern to calculate further terms.

For instance, after 10 hours, you need the 16th term, as 10 hours translate to 15 complete intervals plus the initial one. Use \( a_{16} = 2^{15} \cdot 300 \) to find the number of bacteria, accounting for all 15 doubling events. This method ensures that each calculated term accurately represents the biological growth process.