Question

1. stephanie is collecting data on the reaction of bacteria to different antibodies. she did not measure the area of the bacteria in the dish when she began the experiment but measured 20 cm² of bacteria after 1 hour. she continued to measure the bacteria each hour and entered the data in the table. use stephanies table of data for parts a - c. a. what is the constant ratio in this sequence? b. write an explicit function to represent stephanies data. c. how much area would you predict the bacteria to have when measured during the 5th hour? hours area of bacteria 1 20 2 16 3 12.8 4 10.24 5

Explanation:

Step1: Find the constant ratio

To find the constant ratio $r$ in a geometric - sequence, divide the second term by the first term. Let $a_1 = 20$ (area of bacteria at hour 1) and $a_2 = 16$. Then $r=\frac{a_2}{a_1}=\frac{16}{20}=0.8$.

Step2: Write the explicit function

The general form of an explicit function for a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1$ is the first - term and $r$ is the common ratio. Here, $a_1 = 20$ and $r = 0.8$, so the function is $a_n=20\times(0.8)^{n - 1}$.

Step3: Predict the area at the 5th hour

Substitute $n = 5$ into the explicit function $a_n=20\times(0.8)^{n - 1}$.
\[

$$\begin{align*} a_5&=20\times(0.8)^{5 - 1}\\ &=20\times(0.8)^4\\ &=20\times0.4096\\ & = 8.192 \end{align*}$$

\]

Answer:

A. $0.8$
B. $a_n=20\times(0.8)^{n - 1}$
C. $8.192\ cm^2$