Apply the logistic growth model with the following sets of values to find the population in the next four generations. Be sure to round the population each year to the nearest integer.\

$$N=4000, k=2.5, a_1=1800$$

The breaking strengths B (in tons) of steel cables of various diameters d (in inches) are shown in the following [d: 0.50 B: 9.85] [d: 0.75 B: 21.8] [d: 1.00 B: 38.3] [d: 1.25 B: 59.2] [d: 1.50 B: 84.4] [d: 1.75 B: 114.0]. (a) Use the regression capabilities of a graphing utility to fit an exponential model to the data. (b) Use a graphing utility to plot the data and graph the model. (c) Find the rates of growth of the model when d = 0.8 and d = 1.5.

The logistic growth model $P(t)=\frac{1000}{1+32.33 e^{-0.439 t}}$ represents the population (in grams) of a bacterium after t hours. When will the population be 700 grams?

For the following exercise, use the logistic growth model $f(x)=\frac{150}{1+8 e^{-2 x}}$.
Rewrite $f(x)=1.68(0.65)^x$ as an exponential equation with base $e$ to five significant digits.

The von Bertalanffy growth model is used to predict the length L(t) of a fish over a period of time. If L∞ is the largest length for a species, then the hypothesis is that the rate of growth in length is proportional to L∞ - L, the length yet to be achieved. For the North Sea haddock it has been determined that L∞=53 cm, L(0) = 10 cm, and the constant of proportionality is 0.2. What does the expression for L(t) become with these data?

The logistic growth model represents the population (in grams) of a bacterium after $t$ hours.

$$P(t)=\frac{1000}{1+32.33 e^{-0.439}}$$

Use a graphing utility to graph $P(t)$.

Find a mathematical model representing the statement. (In each case, determine the constant of proportionality.)

$z$ varies directly as the square of $x$ and inversely as $y$. $(z=6$ when $x=6$ and $y=4.)$

The logistic growth model $P(t)=\frac{1000}{1+32.33 e^{-0.439 t}}$ represents the population (in grams) of a bacterium after t hours. Determine the initial population size.

A condo in Austin, Texas, was worth $80,000 in 1990. The value of the condo increased by an average of 3% every year. Give an exponential growth function to model this situation. Then determine the value of the condominium in 2005.

a) Use separation of variables to solve the differential equation model of uninhibited growth,

$$\frac{d P}{d t}=k P .$$

b) Rewrite the solution of part (a) in terms of the condition $P_0=P(0)$.

The logistic growth model $P(t)=\frac{1000}{1+32.33 e^{-0.439 t}}$ represents the population (in grams) of a bacterium after t hours. When will the population be 700 grams?

A model rocket rises from rest with a constant acceleration of 106 m/s$^{2}$. What is the rocket's speed at a height of 3.2 m?

The value of a mountain bike y (in dollars) can be approximated by the model

$$y = 200 ( 0.75 ) ^ { t },$$

where t is the number of years since the bike was new. a. Tell whether the model represents exponential growth or exponential decay. b. Identify the annual percentage increase or decrease in the value of the bike. c. Estimate when the value of the bike will be $50.

For the following exercises, use the logistic growth model $f(x)=\frac{150}{1+8 e^{-2 x}}.$ Rewrite $f(x)=1.68(0.65)^{x}$ as an exponential equation with base e to five significant digits.