91Ó°ÊÓ

Answer each of the following. For the exponential function \(f(x)=a^{x},\) where \(a>1,\) is the function increasing or decreasing over its entire domain?

Decay of Radium Find the half-life of radium- 226 , which decays according to the function \(A(t)=A_{0} e^{-0.00043 t},\) where \(t\) is time in years.

(a) Explain why a polynomial function of even degree cannot have an inverse. (b) Explain why a polynomial function of odd degree may not be one-to-one.

Newton's Law of Cooling A piece of metal is heated to \(300^{\circ} \mathrm{C}\) and then placed in a cooling liquid at \(50^{\circ} \mathrm{C}\). After \(4 \mathrm{min}\), the metal has cooled to \(175^{\circ} \mathrm{C}\). Find its temperature after 12 min. (Hint: Change minutes to hours.)

Population Decline A midwestern city finds its residents moving to the suburbs. Its population is declining according to the function defined by $$ P(t)=P_{0} e^{-0.04 t} $$ where \(t\) is time measured in years and \(P_{0}\) is the population at time \(t=0 .\) Assume that \(P_{0}=1,000,000\) (a) Find the population at time \(t=1\) (b) Estimate the time it will take for the population to decline to \(750,000\). (c) How long will it take for the population to decline to half the initial number?

Growth of Bacteria The growth of bacteria makes it necessary to time-date some food products so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product the number of bacteria present is given by $$ f(t)=500 e^{0.1 t} $$ where \(t\) is time in days and the value of \(f(t)\) is in millions. Find the number of bacteria present at each time. (a) 2 days (b) 4 days (c) 1 week

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. $$\log _{1 / 3} 2$$

Refer to the formulas for compound interest. $$A=P\left(1+\frac{r}{n}\right)^{t n} \text { and } A=P e^{r t} \quad \text { (Section 4.2) }$$ At what interest rate, to the nearest hundredth of a percent, will 16,000 dollars grow to 20,000 dollars if invested for 5.25 yr and interest is compounded quarterly?

Deer Population The exponential growth of the deer population in Massachusetts can be calculated using the model $$ f(x)=50,000(1+0.06)^{x} $$ where \(50,000\) is the initial deer population and 0.06 is the rate of growth. \(f(x)\) is the total population after \(x\) years have passed. (a) Predict the total population after 4 yr. (b) If the initial population was \(30,000\) and the growth rate was \(0.12,\) approximately how many deer would be present after 3 yr? (c) How many additional deer can we expect in 5 yr if the initial population is \(45,000\) and the current growth rate is \(0.08 ?\) (IMAGE CANT COPY)

(Modeling) Solve each problem. See Example 11 . Employee Training A person learning certain skills involving repetition tends to learn quickly at first. Then learning tapers off and skill acquisition approaches some upper limit. Suppose the number of symbols per minute that a person using a keyboard can type is given by $$ f(t)=250-120(2.8)^{-0.5 t} $$ where \(t\) is the number of months the operator has been in training. Find each value. (a) \(f(2)\) (b) \(f(4)\) (c) \(f(10)\) (d) What happens to the number of symbols per minute after several months of training?