Chapter 5

Compound Interest Nancy wants to invest \(\$ 4000\) in saving certificates that bear an interest rate of 9.75\(\%\) per year, compounded semiannully. How long a time period should she choose in order to save an amount of \(\$ 5000 ?\)

\(65-70\) Draw the graph of the function in a suitable viewing rectangle and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=x \log _{10}(x+10) $$

Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model $$ n(t)=\frac{5600}{0.5+27.5 e^{-0.04 t t}} $$ where \(t\) is measured in years. (a) Find the initial bird population. (b) Draw a graph of the function \(n(t)\) (c) What size does the population approach as time goes on?

Doubling an Investment How long will it take for an investment of \(\$ 1000\) to double in value if the interest rate is 8.5\(\%\) per year, compounded continuously?

Compare the rates of growth of the functions \(f(x)=\ln x\) and \(g(x)=\sqrt{x}\) by drawing their graphs on a common screen using the viewing rectangle \([-1,30]\) by \([-1,6]\).

Tree Diameter For a certain type of tree the diameter \(D\) (in feet) depends on the tree's age \(t\) (in years) according to the logistic growth model $$ D(t)=\frac{5.4}{1+2.9 e^{-0.01 t}} $$ Find the diameter of a 20 -year-old tree.

Interest Rate \(\quad\) A sum of \(\$ 1000\) was invested for 4 years, and the interest was compounded semiannully. If this sum amounted to \(\$ 1435.77\) in the given time, what was the interest rate?

Rabbit Population Assume that a population of rabbits behaves according to the logistic growth model $$ n(t)=\frac{300}{0.05+\left(\frac{300}{n_{0}}-0.05\right) e^{-0.55 r}} $$ (a) If the initial population is 50 rabbits, what will the population be after 12 years? (b) Draw graphs of the function \(n(t)\) for \(n_{0}=50,500\) , \(2000,8000\) , and \(12,000\) in the viewing rectangle \([0,15]\) by \([0,12,000] .\) (c) From the graphs in part (b), observe that, regardless of the initial population, the rabbit population seems to approach a certain number as time goes on. What is that number? (This is the number of rabbits that the island can support.)

Annual Percentage Yield Find the annual percentage yield for an investment that earns 8\(\%\) per year, compounded monthly.

\(73-74\) A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .\) (b) How are the graphs in part (a) related? $$ f(x)=\log (c x) $$

Annual Percentage Yield Find the annual percentage yield for an investment that earns 5\(\frac{1}{2} \%\) per year, compounded continuously.

\(73-74\) A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .\) (b) How are the graphs in part (a) related? $$ f(x)=c \log x $$

Radioactive Decay \(\quad\) A 15 -g sample of radioactive iodine decays in such a way that the mass remaining after \(t\) days is given by \(m(t)=15 e^{-0.087 t}\) where \(m(t)\) is measured in grams. After how many days is there only 5 g remaining?

\(75-76\) A function \(f(x)\) is given. (a) Find the domain of the function \(f\) (b) Find the inverse function of \(f .\) $$ f(x)=\log _{2}\left(\log _{10} x\right) $$

Compound Interest If \(\$ 10,000\) is invested at an interest rate of 10\(\%\) per year, compounded semiannully, find the value of the investment after the given number of years. (a) 5 years (b) 10 years (c) 15 years

Skydiving The velocity of a sky diver \(t\) seconds after jumping is given by \(v(t)=80\left(1-e^{-0.2 t}\right) .\) After how many seconds is the velocity 70 \(\mathrm{ft} / \mathrm{s}\) ?

\(75-76\) A function \(f(x)\) is given. (a) Find the domain of the function \(f\) (b) Find the inverse function of \(f .\) $$ f(x)=\ln (\ln (\ln x)) $$

Compound Interest If \(\$ 4000\) is borrowed at a rate of 16\(\%\) interest per year, compounded quarterly, find the amount due at the end of the given number of years. (a) 4 years (b) 6 years (c) 8 years

Fish Population \(\quad\) A small lake is stocked with a certain species of fish. The fish population is modeled by the function $$ P=\frac{10}{1+4 e^{-0.8 t}} $$ where \(P\) is the number of fish in thousands and \(t\) is measured in years since the lake was stocked. (a) Find the fish population after 3 years. (b) After how many years will the fish population reach 5000 fish?

(a) Find the inverse of the function \(f(x)=\frac{2^{x}}{1+2^{x}}\) (b) What is the domain of the inverse function?

Compound Interest If \(\$ 3000\) is invested at an interest rate of 9\(\%\) per year, find the amount of the investment at the end of 5 years for the following compounding methods. (a) Annual (b) Semiannual (c) Monthly (d) Weekly (e) Daily (f) Hourly (g) Continuously

Transparency of a Lake Environmental scientists measure the intensity of light at various depths in a lake to find the "transparency" of the water. Certain levels of transparency are required for the biodiversity of the submerged macrophyte population. In a certain lake the intensity of light at depth \(x\) is given by $$ I=10 e^{-0.008 x} $$ where \(I\) is measured in lumens and \(x\) in feet. (a) Find the intensity \(I\) at a depth of 30 \(\mathrm{ft}\) . (b) At what depth has the light intensity dropped to \(I=5 ?\)

Absorption of Light A spectrophotometer measures the concentration of a sample dissolved in water by shining a light through it and recording the amount of light that emerges. In other words, if we know the amount of light absorbed, we can calculate the concentration of the sample. For a certain substance, the concentration (in moles/liter) is found using the formula $$C=-2500 \ln \left(\frac{I}{I_{0}}\right)$$ where \(I_{0}\) is the intensity of the incident light and \(I\) is the intensity of light that emerges. Find the concentration of the substance if the intensity \(I\) is 70\(\%\) of \(I_{0} .\)

Compound Interest If \(\$ 4000\) is invested in an account for which interest is compounded quarterly, find the amount of the investment at the end of 5 years for the following interest rates. $$ \begin{array}{ll}{\text { (a) } 6 \%} & {\text { (b) } 6 \frac{1}{2} \%} \\\ {\text { (c) } 7 \%} & {\text { (d) } 8 \%}\end{array} $$

Atmospheric Pressure Atmospheric pressure \(P\) (in kilopascals, kPa) at altitude \(h\) (in kilometers, km) is governed by the formula $$ \ln \left(\frac{P}{P_{0}}\right)=-\frac{h}{k} $$ where \(k=7\) and \(P_{0}=100 \mathrm{kPa}\) are constants. (a) Solve the equation for \(P\) . (b) Use part (a) to find the pressure \(P\) at an altude of 4 \(\mathrm{km}\) .

Carbon Dating The age of an ancient artifact can be determined by the amount of radioactive carbon- 14 remaining in it. If \(D_{0}\) is the original amount of carbon- 14 and \(D\) is the amount remaining, then the artifact's age \(A\) (in years) is given by $$A=-8267 \ln \left(\frac{D}{D_{0}}\right)$$ Find the age of an object if the amount \(D\) of carbon- 14 that remains in the object is 73\(\%\) of the original amount \(D_{0} .\)

Cooling an Engine Suppose you're driving your car on a cold winter day \(\left(20^{\circ} \mathrm{F} \text { outside) and the engine overheats (at }\right.\) about \(220^{\circ} \mathrm{F} ) .\) When you park, the engine begins to cool down. The temperature \(T\) of the engine \(t\) minutes after you park satisfies the equation $$ \ln \left(\frac{T-20}{200}\right)=-0.11 t $$ (a) Solve the equation for \(T\) . (b) Use part (a) to find the temperature of the engine after 20 \(\min (t=20) .\)

Bacteria Colony A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time \(t\) (in hours) required for the colony to grow to \(N\) bacteria is given by $$t=3 \frac{\log (N / 50)}{\log 2}$$ Find the time required for the colony to grow to a million bacteria.

Compound Interest Which of the given interest rates and compounding periods would provide the better investment? (i) 9\(\frac{1}{4} \%\) per year, compounded semiannually (ii) 9\(\%\) per year, compounded continuously

Electric Circuits An electric circuit contains a battery that produces a voltage of 60 volts \((\mathrm{V})\) , a resistor with a resistance of 13 ohms \((\Omega),\) and an inductor with an inductance of 5 henrys \((\mathrm{H}),\) as shown in the figure. Using calculus, it can be shown that the current \(I=I(t)\) (in amperes, A) \(t\) seconds after the switch is closed is \(I=\frac{60}{13}\left(1-e^{-13 / 15}\right)\) (a) Use this equation to express the time \(t\) as a function of the current \(I .\) (b) After how many seconds is the current 2 \(\mathrm{A} ?\)

Investment The time required to double the amount of an investment at an interest rate \(r\) compounded continuously is given by $$t=\frac{\ln 2}{r}$$ Find the time required to double an investment at \(6 \%, 7 \%,\) and 8\(\% .\)

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. (a) Find the present value of \(\$ 10,000\) if interest is paid at a rate of 9\(\%\) per year, compounded semiannully, for 3 years. (b) Find the present value of \(\$ 100,000\) if interest is paid at a rate of 8\(\%\) per year, compounded monthly, for 5 years.

Learning Curve A learning curve is a graph of a function \(P(t)\) that measures the performance of someone learning a skill as a function of the training time \(t\) . At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value \(M,\) the rate of learning decreases. It has been found that the function $$ P(t)=M-C e^{-k t} $$ where \(k\) and \(C\) are positive constants and \(C

Charging a Battery The rate at which a battery charges is slower the closer the battery is to its maximum charge \(C_{0}\) . The time (in hours) required to charge a fully discharged battery to a charge \(C\) is given by $$t=-k \ln \left(1-\frac{C}{C_{0}}\right)$$ where \(k\) is a positive constant that depends on the battery. For a certain battery, \(k=0.25 .\) If this battery is fully discharged, how long will it take to charge to 90\(\%\) of its maximum charge \(C_{0} ?\)

Investment \(A\) sum of \(\$ 5000\) is invested at an interest rate of 9\(\%\) per year, compounded semiannully. (a) Find the value \(A(t)\) of the investment after \(t\) years. (b) Draw a graph of \(A(t)\) . (c) Use the graph of \(A(t)\) to determine when this investment will amount to \(\$ 25,000\) .

Estimating a Solution Without actually solving the equation, find two whole numbers between which the solution of \(9^{x}=20\) must lie. Do the same for \(9^{x}=100\) . Explain how you reached your conclusions.

Difficulty of a Task The difficulty in "acquiring a target" (such as using your mouse to click on an icon on your computer screen) depends on the distance to the target and the size of the target. According to Fitts's Law, the index of difficulty (ID) is given by $$\mathrm{ID}=\frac{\log (2 A / W)}{\log 2}$$ where \(W\) is the width of the target and \(A\) is the distance to the center of the target. Compare the difficulty of clicking on an icon that is 5 \(\mathrm{mm}\) wide to one that is 10 \(\mathrm{mm}\) wide. In each case, assume the mouse is 100 \(\mathrm{mm}\) from the icon.

Growth of an Exponential Function Suppose you are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you? (a) One million dollars at the end of the month (b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, \(2^{n}\) cents on the \(n\) th day

A Surprising Equation Take logarithms to show that the equation $$ x^{1 / \log x}=5 $$ has no solution. For what values of \(k\) does the equation $$ x^{1 / \log x}=k $$ have a solution? What does this tell us about the graph of the function \(f(x)=x^{1 / \log x} ?\) Confirm your answer using a graphing device.

The Height of the Graph of a Logarithmic Function Suppose that the graph of \(y=2^{x}\) is drawn on a coordinate plane where the unit of measurement is an inch. (a) Show that at a distance 2 ft to the right of the origin the height of the graph is about 265 \(\mathrm{mi}\) (b) If the graph of \(y=\log _{2} x\) is drawn on the same set of axes, how far to the right of the origin do we have to go before the height of the curve reaches 2 \(\mathrm{ft} ?\)

The Height of the Graph of an Exponential Function Your mathematics instructor asks you to sketch a graph of the exponential function $$ f(x)=2^{x} $$ for \(x\) between 0 and \(40,\) using a scale of 10 units to one inch. What are the dimensions of the sheet of paper you will need to sketch this graph?

Disguised Equations Each of these equations can be transformed into an equation of linear or quadratic type by applying the hint. Solve each equation. $$ \begin{array}{l}{\text { (a) }(x-1)^{\log (x-1)}=100(x-1)} \\ {\text { (b) } \log _{2} x+\log _{4} x+\log _{8} x=11} \\ {\text { (c) } 4^{x}-2^{x+1}=3}\end{array} $$

The Googolplex A googol is \(10^{100},\) and a googolplex is \(10^{\text { googol }} .\) Find \(\log (\log (\operatorname{googol}))\) and \(\quad \log (\log (\log (\text { googolplex })))\)

Comparing Logarithms Which is larger, log \(_{4} 17\) or \(\log _{5} 24 ?\) Explain your reasoning.