Biologists stocked a lake with 500 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 8100. The number of fish doubled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation = KP(1 - R), dP k P(t) dt determine the constant k, and then solve the equation to find an expression for the size of the population after t years. (b) How long will it take for the population to increase to 4050 (half of the carrying capacity)? It will take 3.575 years.

Biologists stocked a lake with 300 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 5000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation dP = kp (1-P). (1-²), kP dt determine the constant k, and then solve the equation to find an expression for the size of the population after t years. k= P(t)= (b) How long will it take for the population to increase to 2500 (half of the carrying capacity)? It will take years.

In terms of the model parameters, state the null hypothesis that, after controlling for sales and roe, ros has no effect on CEO salary. State the alternative that better stock market performance increases a CEO’s salary.

A nation's population (to the nearest million) was 281 million in 2000 and 311 in 2010. It is projected that the population in 2050 will be 439 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Use parts (a) through (f) to use one approach. a. Assume that t = 0 corresponds to 2000 and that the population growth is exponential for the first ten years; that is, between 2000 and 2010, the population is given by P(t) = P(0) e". Estimate the growth rate r using this assumption. = (Round to five decimal places as needed.) b. Write the solution of the logistic equation with the value of r found in part (a). Write any populations in the logistic equation in millions of people. P(t) = Use the projected value P(50) = 439 million to find a value of the carrying capacity K. K= (Type an integer or decimal rounded to the nearest hundredth as needed.) c. According to the logistic model…