Question

Writing Assignment 4 The following application and statements a., b., d., e., and f. can be found Sullivan's Precalculus 9th edition pages 336-337. "A strain of E-coli SC18del-recA718 is placed into a nutrient broth at 30° Celsius and allowed to grow. The data given below are collected where x is for time in hours and y is for population. The theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured." Time (hours), x 2.5 3.5 4.5 4.75 5.255 Population, y 0.175 0.38 0.63 0.76 1.20 The data above is credited to Dr. Polly Lavery, Joliet Junior College. a. Insert a scatter diagram or plot of the given data. a. Select two points from the given data to build an exponential model. Please round your final answer to two decimal places and write your answer in the form of yab. Note: need to show work and explain the step by step process leading to the answer. Furthermore, don't use a calculator to find the exponential model. b. Use a graphing calculator to build an exponential model from the given data. Note: include all the given data in the calculator not just two points. In addition, please explain how you input the data in the calculator and the steps to compute the exponential function/model. d. Insert a graph of the exponential model found in part b. onto the scatter diagram from part a. e. Predict the population when x 6 using the exponential model found in part b. Please show and explain your step by step solution, and please do not insert a table to answer this problem f. Predict the time in hours when b. Please problem. the population will reach 2.1 using the exponential model found in part show and explain your step by step solution, and please do not insert a table to answer this

Answer 1

Using TI-84 Plus calculator.

a) Press [STAT]->"Edit". And enter the data in the two lists.

L1 L2 L31 4.5 4.75 5.255 1.2 175 3B 63 P6 L1(1)=2.

Press [GRAPH] and the Scatter plot appears as below.

b) Select two points (2.5,0.175) and (4.75,0.76). The model is $y=ab^x$. Solving

$0.175=ab^{2.5}\\ 0.76=ab^{4.75}\\ \frac{0.76}{0.175}=b^{4.75-2.5}\\ b=\left (\frac{0.76}{0.175} \right )^{1/2.25}\\ b=1.9206835\\ a=0.275/1.9206835^{2.5}=0.053788964$

The model is ${\color{Blue} y=0.0538 \left ( 1.9207\right )^x}$

c) Using the regression utility in TI-84 Plus, (Press [STAT] -> "CALC" -> ExpReg.

ExPReg a= . E13:39382643 b-1 , 94596 954

The exponential model is ${\color{Blue} y=0.0339 \left ( 1.946\right )^x}$

d) The graph of the exponential model is over-laid on the scatter plot as below.

e) When $x=6$, the population is

$y\left ( 6 \right )=0.0339 \left ( 1.946\right )^6\\ {\color{Blue} y\left ( 6 \right )=1.8410}$

f) When $y=2.1$,

$2.1=0.0339 \left ( 1.946\right )^x\\ x=\frac{\ln \frac{2.1}{0.0339}}{\ln 1.946}\\ {\color{Blue} x=6.1977}$