1.) A biologist is comparing the growth of a population of flies per week to the number of flies a bullfrog will consume per week. She has devised an equation to solve for which day (x) the bullfrog would be able to eat the entire population. The equation is 2^{x} = 3x – 1. However, she has observed that the bullfrog cannot eat more than seven flies in one week.Explain to the biologist how she can solve this on a graph using a system of equations. Identify any possible constraints to the situation.
2.)A baseball player has been improving every season since making it to the big leagues. Below is a table of the runs he has scored. His manager wants to try to determine when he will score 243 runs.
Season |
Runs |
1 |
3 |
2 |
9 |
3 |
27 |
Explain how to create the exponential function that represents his run-scoring abilities. Then explain how to convert this function into a logarithmic function and why this can help the manager answer his question.
3.) Crystal is organizing the seating arrangements for a concert in her local park. The first row’s seats are modeled by the function f(x) = 12. To give everyone a good view, each row must have three more seats than the row before it. The number of additional seats a row has is modeled by the function s(x) = 3(x – 1). Explain to Crystal how to create an equation to predict the number of seats in any row. Describe how to determine the number of seats in the 30th row. Use complete sentences.
4.) Tony has been planting strawberries in a garden in his backyard. Tony started with strawberries in one square foot of garden modeled by the function s(x) = 20. He is allowed to convert more of the garden, and each additional square foot he maintains allows his strawberry plants to produce more by a rate of a(x) = 1.2^{x-1}. Explain to Tony how to create an equation to predict the number of strawberries he can expect based on the number of square feet he maintains. Describe how to determine the number of strawberries he will grow with 15 square feet. Use complete sentences.
5.) John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. Explain to John how to represent his first 20 payments in sigma notation. Then explain how to find the sum of his first 20 payments, using complete sentences. Explain why this series is convergent or divergent.
6.)Nick needs to study a possible link between anti-inflammatory drugs and lung disease. Help Nick design a study that would be appropriate. Explain the type of study, the sampling method, the parameter of interest, and the statistic he could use to estimate that parameter. What conclusions would Nick be able to draw from this study?
7.) A field test for a new exam was given to randomly selected seniors. The exams were graded, and the sample mean and sample standard deviation were calculated. Based on the results, the exam creator claims that on the same exam, nine times out of ten, seniors will have an average score within 4% of 70%. Is the confidence interval at 90%, 95%, or 99%? What is the margin of error? Calculate the confidence interval and explain what it means in terms of the situation.
8.) Two gears are connected and are rotating simultaneously. The smaller gear has a radius of 4 inches, and the larger gear has a radius of 7 inches.
Part 1: What is the angle measure, in degrees and rounded to the nearest tenth, through which the larger gear has rotated when the smaller gear has made one complete rotation?
Part 2: How many rotations will the smaller gear make during one complete rotation of the larger gear?
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