91Ó°ÊÓ

Using a Logistic Differential Equation In Exercises 55 and 56, the logistic differential equation models the growth rate of a population. Use the equation to (a) find the value of \(k,(\) b) find the carrying capacity, (c) graph a slope field using a computer algebra system, and (d) determine the value of \(P\) at which the population growth rate is the greatest. $$\frac{d P}{d t}=0.1 P-0.0004 P^{2}$$

Bacteria Growth The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. (a) Find the initial population. (b) Write an exponential growth model for the bacteria population. Let \(t\) represent the time in hours. (c) Use the model to determine the number of bacteria after 8 hours. (d) After how many hours will the bacteria count be \(25,000 ?\)

Forestry\(The value of a tract of timber is \)V(t)=100,000 e^{0.8} \sqrt{t}\( \)V(t)=100,000 e^{0.8} / \hat{t}\(where \)t\( is the time in years, with \)t=0\( corresponding to 2010 . If money earns interest continuously at 10\)\%\( , then the present value of the timber at any time \)t\( is \)A(t)=V(t) e^{-0.10 t}$ Find the year in which the timber should be harvested to maximize the present value function.

Bacteria Growth At time \(t=0,\) a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4 grams. The maximum weight of the culture is 20 grams. $$\begin{array}{l}{\text { (a) Write a logistic equation that models the weight of the }} \\ {\text { bacterial culture. }} \\ {\text { (b) Find the culture's weight after } 5 \text { hours. }} \\ {\text { (c) When will the culture's weight reach } 18 \text { grams? }}\end{array}$$ $$\begin{array}{l}{\text { (d) Write a logistic differential equation that models the }} \\ {\text { growth rate of the culture's weight. Then repeat part (b) }} \\ {\text { using Euler's Method with a step size of } h=1 . \text { Compare }} \\ {\text { the approximation with the exact answer. }} \\\ {\text { (e) At what time is the culture's weight increasing most }} \\\ {\text { rapidly? Explain. }}\end{array}$$

Slope Field Describe the slope field for a logistic differential equation. Explain your reasoning.

Newton's Law of Cooling When an object is removed from a furnace and placed in an environment with a constant temperature of \(80^{\circ} \mathrm{F},\) its core temperature is \(1500^{\circ} \mathrm{F}\) . One hour after it is removed, the core temperature is \(1120^{\circ} \mathrm{F}\) . (a) Write an equation for the core temperature \(y\) of the object \(t\) thours after it is removed from the furnace. (b) What is the core temperature of the object 6 hours after it is removed from the furnace?

Point of Inflection For any logistic equation, show that the point of inflection occurs at \(y=L / 2\) when the solution starts below the carrying capacity \(L\) .

Sailing Ignoring resistance, a sailboat starting from rest accelerates \((d v / d t)\) at a rate proportional to the difference between the velocities of the wind and the boat. (a) The wind is blowing at 20 knots, and after a half-hour, the boat is moving at 10 knots. Write the velocity \(v\) as a function of time \(t .\) (b) Use the result of part (a) to write the distance traveled by the boat as a function of time.

Determining if a Function Is Homogeneous In Exercises \(69-76,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(x, t y)=t^{n} f(x, y) .\) $$f(x, y)=2 \ln x y$$

Errors and Euler's Method The exact solution of the differential equation \(y^{\prime}=-2 y,\) where \(y(0)=4,\) is \(y=4 e^{-2 x} .\) (a) Use a graphing utility to complete the table, where \(y\) is the exact value of the solution, \(y_{1}\) is the approximate solution using Euler's Method with \(h=0.1, y_{2}\) is the approximate solution using Euler's Method with \(h=0.2, e_{1}\) is the absolute error \(\left|y-y_{1}\right|, e_{2}\) is the absolute error \(\left|y-y_{2}\right|\) and \(r\) is the ratio \(e_{1} / e_{2}\) (b)What can you conclude about the ratio \(r\) as \(h\) changes? (c) Predict the absolute error when \(h=0.05\)