Hallmark project for MATH102.3 Spring 2022 Show all work, or explain in words how you did the problem. You are running an experiment to determine the efcacy of an antibiotic that you helped to develop. At 8 am, you Introduce an antibiotic to a culture of 2400 bacteria. At 12 pm the culture contains 1500 bacteria. 1. Assume that the number of bacteria is decreasing by a constant amount each hour. {T here is a linear relationship between the time passing and the number of bacteria in the culture.) Let mt) be the number of bacteria, and t be the number of hours that have passed since the start of the experiment. a. Find the equation that gives the number of bacteria as a function of the number of hours after the culture was started. b. Sketch a graph of the equation, including the y and the x intercept. c. Interpret the graph. In your paragraphI explaining the meanings of the x and y intercepts, and the slope in the context of this application. Be as complete and precise as possible. If you used this equation to nd the number of bacteria in the culture at 8 pm, what answer would it give you? Does that make sense? Discuss the restrictions you would have to put on the domain of the function to keep you from getting meaningless answers. 2. Now assume that the number of bacteria is decreasing by a percentage each hour. {T here is an exponential relationship between number of hours passed, and the number of bacteria in the culture.) a. Find the equation that gives the number of bacteria as a function of the number of hours after the culture was started. (Round r to two decimal places.) b. Sketch a graph of the function, making sure you include the y intercept, one other point, and the horizontal asymptote in your graph. c. Interpret the graph. In the paragraph, explaining the meanings of the y intercept. the rate at which the number of bacteria is decreasing, and the horizontal asymptote. Be as complete and precise as possible. If you used this equation to nd the number of bacteria in the culture at 8 pm, what answer would it give you? Does that make sense? Discuss the restrictions you would have to put on the domain to keep you from getting meaningless answers. 3. Now assume that you counted the number of bacteria a third limel at 4pm. The number of bacteria returned to the original level the same number of bacteria you started out with. a. Sketch a graph of the three points, and connect them with a smooth graph. b. Find a possible equation for the graph. c. In a paragraph, interpret the graph. In your interpretation, include the meaning of the y intercept, the interval of decrease, the interval of increase, and the vertex in the context of this application. Be as complete and precise as possible. 4. You are presenting the results of the experiment to a committee. You really do believe that the antibiotic is effective. You would like to convince your audience that is the case, because they will decide whether to continue funding your research. Which of the equations will you choose to describe your experiment? Why? Consider an aspect of your presentation that someone might question, and argue your case. 5. You decide to reintroduce the antibiotic to the culture every 8 hours, and you count the bacteria every 4 hours. It turns out that the number of bacteria seems to be cycling between the two original values. a. Make a chart showing your data for 8 measurements. b. Sketch a graph of the 8 readings. Label the horizontal axis with both the number of hours after the start of the experiment, and with the time. c. Find a possible equation for the graph. d. Write a paragraph interpreting the graph. Include the meanings of the high and low pointsI intervals of increase and decrease, and the period. Be as complete and precise as possible. How would this result relate to the other three equations, and how would it effect the argument you made in question 4