The von Bertalanffy growth function $L(t)=L_{\infty}-\left(L_{\infty}-L_{0}\right) e^{-k t}$ where k is a positive constant, models the length L of a fish as a function of t, the age of the fish. This model assumes that the fish has a well-defined length $L_{0}$ at birth (t = 0). Calculate $\lim _{t \rightarrow \infty} L(t).$ How do you interpret the answer?

In the particle under constant acceleration model, we identify the variables and parameters $v_{x i}, v_{x j}, a_x, t$, and $x_f-x_i$. Of the equations in the model, he first does not involve $x_f-x_i$, the second and third do not contain $a_x$, the fourth omits $v_{x j}$, and the last leaves out $t$. So, to complete the set, there should be an equation not involving $v_{x i}$.
(b) Use the equation in part

For the following question,

Consider this scenario: A town has an initial population of $75,000$ . It grows at a constant rate of $2,500$ per year for $5$ years.

What is the output in the year $12$ years from the onset of the model?

The von Bertalanffy growth function $L(t)=L_{\infty}-\left(L_{\infty}-L_{0}\right) e^{-k t}$ where k is a positive constant, models the length L of a fish as a function of t, the age of the fish. Here $L_0$ is the length at birth and $L_{\infty}$ is the final length. Suppose that the mass of a fish of length L is given by $M=a L^{3},$ where a is a positive constant. Calculate the rate of change of mass with respect to age.

Write an exponential growth function to model each situation. Determine the domain and range of each function. Then find the value of the function after the given amount of time. Annual sales for a fast food restaurant are $650,000 and are increasing at a rate of 4% per year; 5 years _____

The table below lists the states with the highest and with the lowest population growth rates. Determine in how many years each event can occur. Use the model $P=P_{0}(1+r)^{x},$ where $P_{0}$ is population from the table, as of July, 2007; x is the number of years after July, 2007, P is the projected population and r is the growth rate. Population of Nevada doubles.

Explain the model of evolution known as punctuated equilibrium.

Write an exponential growth or decay function to model each situation. Graph the function. Then find the value of the function after the given amount of time.

The number of student-athletes at a local high school is 300 and is increasing at a rate of 8 percent per year; 5 years.

What is an equilibrium interest-rate model?

A lake is stocked with 500 fish, and the fish population P begins to increase according to the logistic growth model $P=\frac{10,000}{1+19 e^{-t / 5}}, \quad t \geq 0$ where t is measured in months. (a) Find the number of fish in the lake after 4 months. (b) Use a graphing utility to graph the model. Find the number of months it takes for the population of fish to reach 4000. (c) Does the population have a limit as t increases without bound? Explain your reasoning.

Write an exponential growth or decay function to model each situation. Then find the value of the function after the given amount of time.

A new car is worth $\$ 25,000$, and its value decreases by $15 \%$ each year; 6 years.

The relative growth rate r of a function f measures the rate of change of the function compared to its value at a particular point. It is computed as $r(t)=f^{\prime}(t) / f(t)$. Confirm that the relative growth rate in 1999 $(t=0)$ for the logistic model in Exercise 106 is $r(0)=P^{\prime}(0) / P(0)=0.015$. This means the world's population was growing at $1.5 \%$ per year in 1999.

Although the products of t and D are not constant, the products are close in value. When you use mathematics to model a real-world situation, the functions do not always give exact results. Write an inverse variation equation relating t and D that shows a constant product. Then solve the equation for D.

What is meant by a lognormal model of interest rates?