Question

a scientist is conducting an experiment on two types of bacteria to determine which one 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: y = 4^x bacteria 2: y = 4x^2 in these models, y represents the number of bacteria colonies and x represents the number of hours. based on these models, which type of bacteria is growing faster?

Explanation:

Step1: Analyze the growth models

We have two growth models for bacteria:

• Bacteria 1: \( y = 4^x \)
• Bacteria 2: \( y = 4x^2 \)

To determine which bacteria grows faster, we can compare the two functions for a given value of \( x \) (in hours). Let's choose a value of \( x \) (e.g., \( x = 10 \) hours, since the experiment is for two weeks which is \( 2\times7\times24 = 336 \) hours, but a smaller \( x \) can also be used for comparison).

Step2: Calculate \( y \) for Bacteria 1 at \( x = 10 \)

For Bacteria 1, \( y = 4^x \). Substituting \( x = 10 \):
\( y_1 = 4^{10} = 1048576 \)

Step3: Calculate \( y \) for Bacteria 2 at \( x = 10 \)

For Bacteria 2, \( y = 4x^2 \). Substituting \( x = 10 \):
\( y_2 = 4\times(10)^2 = 4\times100 = 400 \)

Step4: Compare \( y_1 \) and \( y_2 \)

Since \( 1048576>400 \) when \( x = 10 \), and as \( x \) increases, the exponential function \( 4^x \) will grow much faster than the quadratic function \( 4x^2 \) (because exponential functions with base \( > 1 \) grow faster than polynomial functions for large \( x \)).

Answer:

Bacteria 1 (with model \( y = 4^x \)) grows faster.