91Ó°ÊÓ

A logistic growth model has the form ________.

Determine the time necessary for \(P\) dollars to double when it is invested at interest rate \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$r=6.5 \%$$

Complete the table for the radioactive isotope. Amount After 1000 Years \(2 \mathrm{g}\) \(0.4 \mathrm{g}\) Initial Quantity \(10 \mathrm{g}\) \(6.5 \mathrm{g}\) Half-life (years) \(1599\) \(5715\) \(5715\) \(24,100\) Isotope $$^{239} \mathrm{Pu}$$

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$8\left(3^{6-x}\right)=40$$

A laptop computer that costs \(\$ 1150\) new has a book value of \(\$ 550\) after 2 years. (a) Find the linear model \(V=m t+b\) (b) Find the exponential model \(V=a e^{k t}\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.

The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k\) ). (b) How many days should pass before this employee is producing 25 units per day?

Use the One-to-One Property to solve the equation for \(x.\) $$e^{2 x-1}=e^{4}$$

At 8: 30 A.M., a coroner went to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At 9: 00 A.M. the temperature was \(85.7^{\circ} \mathrm{F},\) and at 11: 00 A.M. the temperature was \(82.8^{\circ} \mathrm{F}\). From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula $$t=-10 \ln \frac{T-70}{98.6-70}$$ where \(t\) is the time in hours elapsed since the person died and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. (This formula comes from a general cooling principle called Newton's Law of cooling. It uses the assumptions that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death and that the room temperature was a constant \(70^{\circ} \mathrm{F}\).) Use the formula to estimate the time of death of the person.

Determine whether the statement is true or false. Justify your answer. A logistic growth function will always have an \(x\) -intercept.

A philanthropist deposits 5000 in a trust fund that pays \(7.5 \%\) interest, compounded continuously. The balance will be given to the college from which the philanthropist graduated after the money has earned interest for 50 years. How much will the college receive?