Chapter 5

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 \(I=5 ?\)

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

Atmospheric pressure \(P\) (in kilo-pascals, 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 altitude of 4 \(\mathrm{km}\) .

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 that is absorbed, we can calculate the concentration of the sample. For a certain substance the concentration (in moles per liter) is found by 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 \(I_{0}\) .

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 \(20 \min (t=20) .\)

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 \(D_{0}\) .

An electric circuit contains a battery that produces a voltage of 60 volts (V), a resistor with a resistance of 13 ohms \((\Omega),\) and an inductor with an inductance of 5 henrys (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 \(I=\frac{60}{13}\left(1-e^{-13 / 5}\right) .\) (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 \(\mathrm{A}\) ?

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.

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 \(C< M\) is a reasonable model for learning. (a) Express the learning time \(t\) as a function of the performance level \(P .\) (b) For a pole-vaulter in training, the learning curve is given by $$P(t)=20-14 e^{-0.024 t}$$ where \(P(t)\) is the height he is able to pole-vault after \(t\) months. After how many months of training is he able to vault 12 \(\mathrm{ft}\) ? (c) Draw a graph of the learning curve in part (b).

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 \(8 \% .\)

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.

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 maxi- mum charge \(C_{0} ?\)

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 Fits'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 clicking on one that is 10 \(\mathrm{mm}\) wide. In each case, assume that the mouse is 100 \(\mathrm{mm}\) from the icon.

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

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