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?
A. bacteria 1
B. bacteria 2
Answer (1)
Find the intersection points of the two growth functions by solving 4 x = 4 x 2 .
Analyze the growth rates at different values of x .
Compare the growth rates to determine which bacteria grows faster at different times.
Conclude that Bacteria 1 grows faster based on the exponential nature of its growth function: ba c t er ia 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 is the number of bacteria colonies and x is the number of hours. We want to determine which bacteria grows faster.
Finding Intersection Points To compare the growth rates, we can analyze the behavior of the two functions. Let's find the points where the number of colonies is the same for both bacteria types. This means solving the equation 4 x = 4 x 2 . Dividing both sides by 4, we get 4 x − 1 = x 2 . Taking the square root of both sides, we have 2 x − 1 = ∣ x ∣ .
Calculating Roots We can find the intersection points by approximating the roots of the equation 4 x − 4 x 2 = 0 . Using a tool, we find the roots to be approximately x = − 0.383 , x = 1 , and x = 2 . Since x represents the number of hours, we only consider non-negative values. Thus, we have intersection points at x = 1 and x = 2 .
Comparing Growth Rates Let's analyze the growth rates at different values of x :
At x = 0 : Bacteria 1 has y = 4 0 = 1 colony, and Bacteria 2 has y = 4 ( 0 ) 2 = 0 colonies. Bacteria 1 starts with more colonies.
At x = 0.5 : Bacteria 1 has y = 4 0.5 = 2 colonies, and Bacteria 2 has y = 4 ( 0.5 ) 2 = 1 colony. Bacteria 1 has more colonies.
At x = 1 : Bacteria 1 has y = 4 1 = 4 colonies, and Bacteria 2 has y = 4 ( 1 ) 2 = 4 colonies. They have the same number of colonies.
At x = 1.5 : Bacteria 1 has y = 4 1.5 = 8 colonies, and Bacteria 2 has y = 4 ( 1.5 ) 2 = 9 colonies. Bacteria 2 has more colonies.
At x = 2 : Bacteria 1 has y = 4 2 = 16 colonies, and Bacteria 2 has y = 4 ( 2 ) 2 = 16 colonies. They have the same number of colonies.
At x = 2.5 : Bacteria 1 has y = 4 2.5 = 32 colonies, and Bacteria 2 has y = 4 ( 2.5 ) 2 = 25 colonies. Bacteria 1 has more colonies.
At x = 3 : Bacteria 1 has y = 4 3 = 64 colonies, and Bacteria 2 has y = 4 ( 3 ) 2 = 36 colonies. Bacteria 1 has more colonies.
At x = 4 : Bacteria 1 has y = 4 4 = 256 colonies, and Bacteria 2 has y = 4 ( 4 ) 2 = 64 colonies. Bacteria 1 has more colonies.
Determining Faster Growth From the analysis, we can see that initially, Bacteria 1 has a higher growth rate. However, Bacteria 2 catches up and has a higher growth rate between x = 1 and x = 2 . After x = 2 , Bacteria 1's growth rate surpasses Bacteria 2's and continues to grow faster. Since the exponential function 4 x will eventually outpace the quadratic function 4 x 2 , Bacteria 1 will eventually grow faster.
Final Answer Based on the models, Bacteria 1 is growing faster.
Examples
Understanding bacterial growth is crucial in many fields, such as medicine and environmental science. For instance, in medicine, knowing the growth rate of bacteria helps doctors determine the appropriate dosage of antibiotics. If Bacteria 1 represents a harmful bacteria and Bacteria 2 represents a beneficial bacteria, this model can help predict when Bacteria 1 will outgrow Bacteria 2 in a certain environment, informing strategies to control the harmful bacteria's growth. This type of modeling is also used in environmental science to predict the spread of invasive species or the growth of algae in a lake.
