Chapter 5

A drug is eliminated from the body through urine. Suppose that for a dose of 10 milligrams, the amount \(A(t)\) remaining in the body \(t\) hours later is given by \(A(t)=10(0.8)^{t}\) and that in order for the drug to be effective, at least 2 milligrams must be in the body. (a) Determine when 2 milligrams is left in the body. (b) What is the half-life of the drug?

The radioactive bismuth isotope \({ }^{210} \mathrm{Bi}\) disintegrates according to \(Q=k(2)^{-t / 5}\), where \(k\) is a constant and \(t\) is the time in days. Express \(t\) in terms of \(Q\) and \(k\).

The basic source of genetic diversity is mutation, or changes in the chemical structure of genes. If a gene mutates at a constant rate \(m\) and if other evolutionary forces are negligible, then the frequency \(F\) of the original gene after \(t\) generations is given by \(F=F_{0}(1-m)^{t}\), where \(F_{0}\) is the frequency at \(t=0\). (a) Solve the equation for \(t\) using common logarithms. (b) If \(m=5 \times 10^{-5}\), after how many generations does \(F=\frac{1}{2} F_{0}\) ?

Inflation comparisons In 1974 , Johnny Miller won 8 tournaments on the PGA tour and accumulated \(\$ 353,022\) in official season earnings. In 1999 , Tiger Woods accumulated \(\$ 6,616,585\) with a similar record. (a) Suppose the monthly inflation rate from 1974 to 1999 was \(0.0025(3 \% / \mathrm{yr})\). Use the compound interest formula to estimate the equivalent value of Miller's winnings in the year 1999 . Compare your answer with that from an inflation calculation on the web (e.g., bls.gov/cpi/home.htm). (b) Find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings. (c) What type of function did you use in part (a)? part (b)?

The energy \(E(x)\) of an electron after passing through material of thickness \(x\) is given by the equation \(E(x)=E_{0} e^{-x / x_{0}}\), where \(E_{0}\) is the initial energy and \(x_{0}\) is the radiation length. (a) Express, in terms of \(E_{0}\), the energy of an electron after it passes through material of thickness \(x_{0}\). (b) Express, in terms of \(x_{0}\), the thickness at which the electron loses \(99 \%\) of its initial energy.

Certain learning processes may be illustrated by the graph of an equation of the form \(f(x)=a+b\left(1-e^{-c}\right)\), where \(a, b\), and \(c\) are positive constants. Suppose a manufacturer estimates that a new employee can produce five items the first day on the job. As the employee becomes more proficient, the daily production increases until a certain maximum production is reached. Suppose that on the \(n\)th day on the job, the number \(f(n)\) of items produced is approximated by $$ f(n)=3+20\left(1-e^{-0.1 n}\right) . $$ (a) Estimate the number of items produced on the fifth day, the ninth day, the twenty-fourth day, and the thirtieth day. (b) Sketch the graph of \(f\) from \(n=0\) to \(n=30\). (Graphs of this type are called learning curves and are used frequently in education and psychology.) (c) What happens as \(n\) increases without bound?

An electrical condenser with initial charge \(Q_{0}\) is allowed to discharge. After \(t\) seconds the charge \(Q\) is \(Q=Q_{0} e^{k t}\), where \(k\) is a constant. Solve this equation for \(t\).

One method of estimating the thickness of the ozone layer is to use the formula $$ \ln I_{0}-\ln I=k x, $$ where \(I_{0}\) is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, \(I\) is the intensity of the same wavelength after passing through a layer of ozone \(x\) centimeters thick, and \(k\) is the absorption constant of ozone for that wavelength. Suppose for a wavelength of \(3176 \times 10^{-8} \mathrm{~cm}\) with \(k \approx 0.39, I_{0} / I\) is measured as 1.12. Approximate the thickness of the ozone layer to the nearest \(0.01\) centimeter.

The growth in height of trees is frequently described by a logistic equation. Suppose the height \(h\) (in feet) of a tree at age \(t\) (in years) is $$ h=\frac{120}{1+200 e^{-0.2 t}}, $$ as illustrated by the graph in the figure. (a) What is the height of the tree at age 10? (b) At what age is the height 50 feet?

Use the Richter scale formula \(R=\log \left(I / I_{0}\right)\) to find the magnitude of an earthquake that has an intensity (a) 100 times that of \(I_{0}\) (b) 10,000 times that of \(I_{0}\) (c) 100,000 times that of \(I_{0}\)

Manufacturers sometimes use empirically based formulas to predict the time required to produce the \(n\)th item on an assembly line for an integer \(n\). If \(T(n)\) denotes the time required to assemble the \(n\)th item and \(T_{1}\) denotes the time required for the first, or prototype, item, then typically \(T(n)=T_{1} n^{-k}\) for some positive constant \(k\). (a) For many airplanes, the time required to assemble the second airplane, \(T(2)\), is equal to \((0.80) T_{1}\). Find the value of \(k\). (b) Express, in terms of \(T_{1}\), the time required to assemble the fourth airplane. (c) Express, in terms of \(T(n)\), the time \(T(2 n)\) required to assemble the \((2 n)\) th airplane.

Refer to Exercises 67-68 in Section 3.3. If \(v_{0}\) is the wind speed at height \(h_{0}\) and if \(v_{1}\) is the wind speed at height \(h_{1}\), then the vertical wind shear can be described by the equation $$ \frac{v_{0}}{v_{1}}=\left(\frac{h_{0}}{h_{1}}\right)^{P} $$ where \(P\) is a constant. During a one-year period in Montreal, the maximum vertical wind shear occurred when the winds at the 200 -foot level were \(25 \mathrm{mi} / \mathrm{hr}\) while the winds at the 35 -foot level were \(6 \mathrm{mi} / \mathrm{hr}\). Find \(P\) for these conditions.

The loudness of a sound, as experienced by the human ear, is based on its intensity level. A formula used for finding the intensity level \(\alpha\) (in decibels) that corresponds to a sound intensity \(I\) is \(\alpha=10 \log \left(I / I_{0}\right)\), where \(I_{0}\) is a special value of \(I\) agreed to be the weakest sound that can be detected by the ear under certain conditions. Find \(\alpha\) if (a) \(I\) is 10 times as great as \(I_{0}\) (b) \(I\) is 1000 times as great as \(I_{0}\) (c) \(I\) is 10,000 times as great as \(I_{0}\) (This is the intensity level of the average voice.)

Exer. 63-64: An economist suspects that the following data points lie on the graph of \(y=c 2^{k x}\), where \(c\) and \(k\) are constants. If the data points have three-decimal-place accuracy, is this suspicion correct? $$ (0,4),(1,3.249),(2,2.639),(3,2.144) $$

The population \(N(t)\) (in millions) of the United States \(t\) years after 1980 may be approximated by the formula \(N(t)=231 e^{0.0103 t}\). When will the population be twice what it was in 1980 ?

Exer. 63-64: An economist suspects that the following data points lie on the graph of \(y=c 2^{k x}\), where \(c\) and \(k\) are constants. If the data points have three-decimal-place accuracy, is this suspicion correct? $$ \begin{aligned} &(0,-0.3),(0.5,-0.345),(1,-0.397),(1.5,-0.551) \\ &(2,-0.727) \end{aligned} $$

The population \(N(t)\) (in millions) of India \(t\) years after 1985 may be approximated by the formula \(N(t)=766 e^{0.0182 t}\). When will the population reach \(1.5\) billion?

The Ehrenberg relation $$ \ln W=\ln 2.4+(1.84) h $$ is an empirically based formula relating the height \(h\) (in meters) to the average weight \(W\) (in kilograms) for children 5 through 13 years old. (a) Express \(W\) as a function of \(h\) that does not contain \(\ln\). (b) Estimate the average weight of an 8-year-old child who is \(1.5\) meters tall.

Exer. 65-66: It is suspected that the following data points lie on the graph of \(y=c \log (k x+10)\), where \(c\) and \(k\) are constants. If the data points have three-decimal-place accuracy, is this suspicion correct? $$ (0,0.7),(1,0.782),(2,0.847),(3,0.900),(4,0.945) $$

t If interest is compounded continuously at the rate of 6% per year, approximate the number of years it will take an initial deposit of \(6000 to grow to \)25,000.

Exer. 67-68: Approximate the function at the value of \(x\) to four decimal places. $$ h(x)=\log _{4} x-2 \log _{8} 1.2 x ; \quad x=5.3 $$

Exer. 67-68: Approximate the function at the value of \(x\) to four decimal places. $$ h(x)=3 \log _{3}(2 x-1)+7 \log _{2}(x+0.2) ; \quad x=52.6 $$

A liquid's vapor pressure \(P\) (in \(\mathrm{lb} / \mathrm{in}^{2}\) ), a measure of its volatility, is related to its temperature \(T\) (in \({ }^{\circ} \mathrm{F}\) ) by the Antoine equation $$ \log P=a+\frac{b}{c+T} $$ where \(a, b\), and \(c\) are constants. Vapor pressure increases rapidly with an increase in temperature. Express \(P\) as a function of \(T\).

The weight \(W\) (in kilograms) of a female African elephant at age \(t\) (in years) may be approximated by $$ W=2600\left(1-0.51 e^{-0.075 t}\right)^{3} . $$ (a) Approximate the weight at birth. (b) Estimate the age of a female African elephant weighing 1800 kilograms by using (1) the accompanying graph and (2) the formula for \(W\).

A jar of boiling water at \(212^{\circ} \mathrm{F}\) is set on a table in a room with a temperature of \(72^{\circ} \mathrm{F}\). If \(T(t)\) represents the temperature of the water after \(t\) hours, determine which function best models the situation. (1) \(T(t)=212-50 t\) (2) \(T(t)=140 e^{-t}+72\) (3) \(T(t)=212 e^{-t}\) (4) \(T(t)=72+10 \ln (140 t+1)\)

Urban population density An urban density model is a formula that relates the population density \(D\) (in thousands/ \(\mathrm{mi}^{2}\) ) to the distance \(x\) (in miles) from the center of the city. The formula \(D=a e^{-b x}\) for the central density \(a\) and coefficient of decay \(b\) has been found to be appropriate for many large U.S. cities. For the city of Atlanta in \(1970, a=5.5\) and \(b=0.10\). At approximately what distance was the population density 2000 per square mile?

Stars are classified into categories of brightness called magnitudes. The faintest stars, with light flux \(L_{0}\), are assigned a magnitude of 6 . Brighter stars of light flux \(L\) are assigned a magnitude \(m\) by means of the formula $$ m=6-2.5 \log \frac{L}{L_{0}} $$ (a) Find \(m\) if \(L=10^{0.4} L_{0}\). (b) Solve the formula for \(L\) in terms of \(m\) and \(L_{0}\).

Radioactive iodine \({ }^{131} \mathrm{I}\) is frequently used in tracer studies involving the thyroid gland. The substance decays according to the formula \(A(t)=A_{0} a^{-t}\), where \(A_{0}\) is the initial dose and \(t\) is the time in days. Find \(a\), assuming the half-life of \({ }^{131} \mathrm{I}\) is 8 days.

In a survey of 15 cities ranging in population \(P\) from 300 to \(3,000,000\), it was found that the average walking speed \(S\) (in \(\mathrm{ft} / \mathrm{sec})\) of a pedestrian could be approximated by \(S=0.05+0.86 \log P\). (a) How does the population affect the average walking speed? (b) For what population is the average walking speed \(5 \mathrm{ft} / \mathrm{sec}\) ?

For manufacturers of computer chips, it is important to consider the fraction \(F\) of chips that will fail after \(t\) years of service. This fraction can sometimes be approximated by the formula \(F=1-e^{-c t}\), where \(c\) is a positive constant. (a) How does the value of \(c\) affect the reliability of a chip? (b) If \(c=0.125\), after how many years will \(35 \%\) of the chips have failed?

(a) \(f(x)=\ln (x+1)+e^{x}, \quad x=2\) (b) \(g(x)=\frac{(\log x)^{2}-\log x}{4}, x=3.97\)

(a) \(f(x)=\log \left(2 x^{2}+1\right)-10^{-x}, \quad x=1.95\) (b) \(g(x)=\frac{x-3.4}{\ln x+4}\), \(x=0.55\)

Studies relating serum cholesterol level to coronary heart disease suggest that a risk factor is the ratio \(x\) of the total amount \(C\) of cholesterol in the blood to the amount \(H\) of high-density lipoprotein cholesterol in the blood. For a female, the lifetime risk \(R\) of having a heart attack can be approximated by the formula $$ R=2.07 \ln x-2.04 \text { provided } 0 \leq R \leq 1 . $$ For example, if \(R=0.65\), then there is a \(65 \%\) chance that a woman will have a heart attack over an average lifetime. Calculate \(R\) for a female with \(C=242\) and \(H=78\).