A scientist is conducting an experiment on two types of bacteria to determine which type will grow faster in a pool. After collecting data for two weeks, she finds that she can model the growth rates as follows:
Bacteria 1: [tex]y=4^x[/tex]
Bacteria 2: [tex]y=4 x^2[/tex]
In these models, [tex]y[/tex] represents the number of bacteria colonies and [tex]x[/tex] represents the number of hours.
Based on these models, which type of bacteria is growing faster?
How do you know which bacteria is growing faster?
A. An exponential function grows faster than a linear function.
B. An exponential function grows faster than a quadratic function.
C. A quadratic function grows faster than an exponential function.
Answer (1)
Bacteria 1 grows exponentially, modeled by y = 4 x .
Bacteria 2 grows quadratically, modeled by y = 4 x 2 .
Exponential functions grow faster than quadratic functions as x increases.
Therefore, Bacteria 1 grows faster. $\boxed{\text{Bacteria 1}}
Explanation
Understanding the Problem We are given two models for bacteria growth:
Bacteria 1: y = 4 x Bacteria 2: y = 4 x 2
where y represents the number of bacteria colonies and x represents the number of hours. We need to determine which bacteria type grows faster.
Identifying Growth Models Bacteria 1 has an exponential growth model, while Bacteria 2 has a quadratic growth model. Exponential functions eventually grow much faster than polynomial functions (like quadratic functions) as x increases.
Comparing Growth Rates To illustrate this, let's compare the growth rates at different values of x :
At x = 1 : Bacteria 1 = 4 1 = 4 , Bacteria 2 = 4 ( 1 ) 2 = 4 At x = 2 : Bacteria 1 = 4 2 = 16 , Bacteria 2 = 4 ( 2 ) 2 = 16 At x = 3 : Bacteria 1 = 4 3 = 64 , Bacteria 2 = 4 ( 3 ) 2 = 36 At x = 4 : Bacteria 1 = 4 4 = 256 , Bacteria 2 = 4 ( 4 ) 2 = 64 At x = 5 : Bacteria 1 = 4 5 = 1024 , Bacteria 2 = 4 ( 5 ) 2 = 100
As we can see, after a certain point, Bacteria 1 starts growing much faster than Bacteria 2.
Conclusion Since exponential functions grow faster than quadratic functions in the long run, Bacteria 1 will grow faster.
Examples
Imagine you're investing money. One investment grows exponentially (like Bacteria 1), and another grows quadratically (like Bacteria 2). Initially, they might grow similarly, but over time, the exponential investment will significantly outperform the quadratic one. This principle applies to various scenarios, such as population growth, compound interest, and the spread of information, where exponential growth can lead to dramatic changes compared to polynomial growth.
