Solve each problem (Modeling) Bacterial Growth Suppose that...

Solve each problem (Modeling) Bacterial Growth Suppose that a strain of bacteria will double in size and then divide every 40 minutes. Let $a_{1}$ be the initial number of bacteria cells, $a_{2}$ the number after 40 minutes, and $a_{n}$ the number after $40(n-1)$ minutes. (image can't copy) (a) Write a formula for the $n$ th term $a_{n}$ of the geometric sequence $$ a_{1}, a_{2}, a_{3}, \ldots, a_{m}, \ldots $$ (b) Determine the first value for $n$ where $a_{m}>1,000,000$ if $a_{1}=100$ (c) How long does it take for the number of bacteria to exceed one million?

Solve each problem
(Modeling) Bacterial Growth Suppose that a strain of bacteria will double in size and then divide every 40 minutes. Let $a_{1}$ be the initial number of bacteria cells, $a_{2}$ the number after 40 minutes, and $a_{n}$ the number after $40(n-1)$ minutes.
(image can't copy)
(a) Write a formula for the $n$ th term $a_{n}$ of the geometric sequence
$$ a_{1}, a_{2}, a_{3}, \ldots, a_{m}, \ldots $$
(b) Determine the first value for $n$ where $a_{m}>1,000,000$ if $a_{1}=100$
(c) How long does it take for the number of bacteria to exceed one million?

Key Concepts

Geometric Sequences

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant called the common ratio. This concept is fundamental in modeling situations where a quantity changes by a fixed percentage or factor over equal time intervals, allowing one to describe the nth term with the formula: a? = a? · r^(n-1).

Exponential Growth Modeling

Exponential growth refers to the process by which a quantity increases by a constant multiplicative rate over equal time intervals. In many real-world applications, such as bacterial population growth, the amount doubles or multiplies by a fixed factor at periodic intervals, leading to rapid increases that can be effectively modeled using exponential functions.

Solving Exponential Inequalities

Solving exponential inequalities involves finding the point at which an exponentially growing quantity exceeds a specified threshold. The process typically requires setting up an inequality using the exponential formula and solving for the exponent, often using logarithms or iterative estimation, which is essential in determining when a modeled event occurs.

Time Unit Conversion

Time unit conversion involves translating a count of discrete time intervals into an actual measure of time, such as minutes or hours. When models involve events happening at regular intervals, converting the number of intervals into total elapsed time is crucial for relating abstract numerical indices to real-world time measurements.

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