Reminder Round all answers to two decimal places unless otherwise indicated.
Dispersion Model Animal populations move about and disperse. A number of models for this dispersion have been proposed, and many of them involve the logarithm. For example, in
a. Make a graph of
b. How many pill bugs were to be found within
c. How far from the release point would you expect to find only a single individual?
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FUNCTIONS AND CHANGE COMBO
- Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 4 shows the number of games sold, in thousands, from the years 20002010. a. Let x represent time in years starting with x=1 for the year 2000. Let y represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data. b. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.arrow_forwardWhat might a scatterplot of data points look like if it were best described by a logarithmic model?arrow_forwardLogistic Population growth the table and scatter plot give the population of black flies in a closed laboratory container over an 18 day period. (a) Use the logistic command on your calculator to find a logistic model for these data. (b) Use the model to estimate the time when there were 400 flies in the containerarrow_forward
- What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.arrow_forwardWhat does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?arrow_forwardMore on the Pacific Sardine This is a continuation of Example 5.1. In this exercise, we explore the Pacific sardine population further, using the model in Example 5.1. a. If the current level of the Pacific sardine population is 50,000 tons, how long will it take for the population to recover to the optimum growth level of 1.2milliontons? Suggestion: One way to solve this is to make a new logistic formula using K2.4, r0.338, and N(0)0.05. b. The value of r used in Example 5.1 ignores the effects of fishing. If fishing mortality is taken into account, then r drops to 0.215 per year with the carrying capacity still at 2.4milliontons. Answer the question in part a using this lower value of r. Note: The population estimate of 50,000 tons and the adjusted value of r are given in the paper by Murphy see footnote 3 on page 347. Murphy points out that factoring in the growth of the competing anchovy population makes the recovery times even longer, and he adds. "It is disconcerting to realize how slowly the population will recover to its level of maximum productivity ... even if fishing stops." Studies to fit a logistic model to the Pacific sardine population have yielded. N=241+239e0.338t where t is measured in years and N is measured in millions of tons of fish. Part 1 What is r for the Pacific sardine? Part 2 According to the logistic model, in the absence of limiting factors, what would be the annual percentage growth rate for the Pacific sardine? Part 3 What is the environmental carrying capacity K? Part 4 What is the optimum yield level? Part 5 Make a graph of N versus t. Part 6 At what time t should the population he harvested? Part 7 What portion of the graph is concave up? What portion is concave down?arrow_forward
- The Decibel scale Exercise S-7 through S-10 refer to the decibel scale. If one sound has a relative intensity one-tenth that of another, how do their decibel levels compare?arrow_forwardTorontos Jewish Population The table gives the population of Torontos Jewish community at various times. Source: The Mathematics Teacher. Plot the population on the y-axis against the year on the x-axis. Let x represents the years since 1900. Do the data appear to lie along a straight line? Plot the natural logarithm of the population against the year. Does the graph appear to be more linear than the graph in part a? Find an equation for the least squares line for the data plotted in part b. If your graphing calculator has an exponential regression feature, find the exponential function that best fits the given data according to the least squares method. Take the natural logarithm of the equation found in part d, and verify that the result is same as the equation found in part c. In Section 11.1 on Solutions of Elementary and Separable Differential Equations, we will see another type of function that is a better fit to these data.arrow_forward
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