Gray wolves recolonized in the Upper Peninsula of Michigan beginning in 1990.

Gray wolves recolonized in the Upper Peninsula of Michigan beginning in 1990.

Their population has been documented as shown in the table below.
Gray Wolf Population in the Upper Peninsula of Michigan
1990 1992 1994 1996 1998 2000
6 21 57 116 140 216
You assumed that a linear model was appropriate and found the equation for the best-fit linear.
(a) Using your calculator, find the equation for the linear model. Let Wet) be the number
of wolves in the year “t”. Let ”t” be defined as the number of years since 1990.
(b) Calculate (by hand) the residuals. Show all work.
(c) Create a residual plot. Plot it by hand using graph paper. (Attach the graph to the end of the assignment.) Reminder: Make sure that the graph has a title and the axes are labeled.
(d)Do you feel that you made a good choice for the model? Explain by referring to the characteristics of the residual plot.
2) The half-life of radioactive Uranium II (U234) is about 250,000 years. What percent of radioactive uranium will remain after 150,000 years? Show all calculations.
3) A species of bat is in danger of becoming extinct. Five years ago, the total population of the species was 2000. Two years ago, the total population was 1400. Assume that the population is declining exponentially.
(a) Find an exponential function b(t) that fits the data. Let b(t) be the number of bats at time “t” and let “t” be the number of years after the count began five years ago. Show all work.
(b) Using the exponential model from (a), find the population of bats a year ago. Show all work.
(c) Estimate the length of time to extinction. Show calculations. ( Explain the assumptions that you needed to make in order to calculate the time to extinction.)
4) The body eliminates 17% of the caffeine present each hour. You just consumed 36 ounces of coffee (approximately 520 mg. of caffeine) and a can of Mountain Dew (approximately 55 mg. of caffeine.
(a) Write the equation for the exponential decay for the amount of caffeine left in the body after each period of one hour. Use C(t) to represent the amount of caffeine in the body at time “t”. Define “t” as the number of hours after the initial consumption of the caffeine.
(b) After three hours, how much caffeine remains in your body? Show your calculations.
(c) Most people tolerate up to 200 mg of caffeine before they become ‘jittery”. How long will it take before your caffeine level gets below this benchmark? Show the calculations.
5) Sixty percent of a radioactive substance decays in 15 years. By how much does the substance
decay each year? Show all work.
6. Find the annual growth rate if the quantity grows by 7% in 8 months. Show all work.
7. The Park Ranger finds a black bear dead in the park at 7 am. The bear had been shot, probably over night. To help find the perpetrator of the killing, the ranger wants to determine the time of the shooting. He records the body temperature of the bear in half hour intervals. The table below is the record of the times that the temperature was taken and the difference between the body temperature and the air temperature at each of those times. The ranger recorded the times as “t”, where t = the number of half hour periods after 7 am. Thus t = 0 means 7 am and t = 1 means 7:30 am. “D” is the difference in temperature between the bear carcass and the air ~ temperature (body temp – air temp). The bear was found in a thickly wooded (shaded) area andthe skies had been thickly overcast all evening and throughout the day that the bear carcass was discovered. This allowed very little radiational cooling over night and little warming during the day. So, we may assume that the air temperature remained nearly constant at 65 degrees F. You will need to know that the average body temperature of black bears is about 95 degrees F.
t 0 1 2 3 4 5 6 7 8 9 10
D 13.5 12.25 10.75 10 9 8.25 7.5 6.75 6 5.5 5

(a) Provide an argument to support the use of an exponential model. Show your calculations and comment on the results.
(b) What is the decay factor for each half-hour period? You may round your answer to one decimal place.
(c) What is the decay factor for each one hour period? Give the answer to two decimal places.
(d) Find an exponential model for the data that shows temperature difference as a function of hours. Use the notation D(x) as the difference in temperature and x as the number of one hours periods that have elapsed since the bear carcass was discovered.
(e) Find a formula for the function T = T(x) that gives the temperature of bear at time x.
(f) Find the time of death of the bear. Show all calculations.

 
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