Chapter 1

To measure temperature, three scales are commonly used: Fahrenheit, Celsius, and Kelvin. These scales are linearly related. (a) The Kelvin (K) scale is an absolute scale of temperature. The zero point of the scale \((0 \mathrm{~K})\) denotes absolute zero, the coldest possible temperature; that is, no body can have a temperature below \(0 \mathrm{~K}\). It has been determined experimentally that \(0 \mathrm{~K}\) corresponds to \(-273.15^{\circ} \mathrm{C}\). If \(1 \mathrm{~K}\) denotes the same temperature difference as \(1^{\circ} \mathrm{C}\), find an equation that relates the Kelvin and Celsius scales. (b) Pure nitrogen and pure oxygen can be produced cheaply by first liquefying purified air and then allowing the temperature of the liquid air to rise slowly. Since nitrogen and oxygen have different boiling points, they are distilled at different temperatures. The boiling point of nitrogen is \(77.4 \mathrm{~K}\) and of oxygen is \(90.2 \mathrm{~K}\). Convert each of these boiling-point temperatures into Celsius. If you solved Problem \(52(\) a), convert the boiling-point temperatures into Fahrenheit as well. Consider the two techniques described for distilling nitrogen and oxygen. Which element gets distilled first?

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=5^{-6 x} $$

Use the following steps to show that if two nonvertical lines \(l_{1}\) and \(l_{2}\) with slopes \(m_{1}\) and \(m_{2}\), respectively, are perpendicular, then \(m_{1} m_{2}=-1\) : Assume that \(m_{1}<0\) and \(m_{2}>0\). (a) Use a graph to show that if \(\theta_{1}\) and \(\theta_{2}\) are the respective angles of inclination of the lines \(l_{1}\) and \(l_{2}\), then \(\theta_{1}=\theta_{2}+\frac{\pi}{2} .\) (The angle of inclination of a line is the angle \(\theta \in[0, \pi)\) between the line and the positively directed \(x\) -axis.) (b) Use the fact that \(\tan (\pi-x)=-\tan x\) to show that \(m_{1}=\) \(\tan \theta_{1}\) and \(m_{2}=\tan \theta_{2}\) (c) Use the fact that \(\tan \left(\frac{\pi}{2}-x\right)=\cot x\) and \(\cot (-x)=-\cot x\) to show that \(m_{1}=-\cot \theta_{2}\) (d) From the latter equation, deduce the truth of the claim set forth at the beginning of this problem.

Find the equation of a circle with center \((-1,4)\) and radius \(3 .\)

When \(\log y\) is graphed as a function of \(\log x, a\) straight line results. Graph straight lines, each given by two points, 5n a log-log plot, and determine the functional relationship. $$ \left(x_{1}, y_{1}\right)=(3,5),\left(x_{2}, y_{2}\right)=(1,5) $$

Sketch each scaling relation (Niklas, 1994). Suppose that a cube of length \(L\) and volume \(V\) has mass \(M\) and that \(M=0.35 V\). How does the length of the cube depend on its mass?

Find the equation of a circle with center \((2,3)\) and radius 4 .

When \(\log y\) is graphed as a function of \(\log x, a\) straight line results. Graph straight lines, each given by two points, 5n a log-log plot, and determine the functional relationship. $$ \left(x_{1}, y_{1}\right)=(4,2),\left(x_{2}, y_{2}\right)=(8,8) $$

Assume that a population size at time \(t\) is \(N(t)\) and that $$ N(t)=2^{t}, \quad t \geq 0 $$ (a) Find the population size for \(t=0,1,2,3\), and 4 . (b) Graph \(N(t)\) for \(t \geq 0\).

(a) Find the equation of a circle with center \((2,5)\) and radius \(3 .\) (b) Where does the circle intersect the \(y\) -axis? (c) Does the circle intersect the \(x\) -axis? Explain.

When \(\log y\) is graphed as a function of \(\log x, a\) straight line results. Graph straight lines, each given by two points, 5n a log-log plot, and determine the functional relationship. $$ \left(x_{1}, y_{1}\right)=(2,5),\left(x_{2}, y_{2}\right)=(5,2) $$

Assume that a population size at time \(t\) is \(N(t)\) and that $$ N(t)=40 \cdot 2^{t}, \quad t \geq 0 $$ (a) Find the population size at time \(t=0\). (b) Show that $$ N(t)=40 e^{t \ln 2}, \quad t \geq 0 $$

(a) Find all possible radii of a circle centered at \((3,6)\) so that the circle intersects only one axis. (b) Find all possible radii of a circle centered at \((3,6)\) so that the circle intersects both axes.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=2 x^{5} $$

The half-life of \(\mathrm{C}^{14}\) is 5730 years. If a sample of \(\mathrm{C}^{14}\) has a mass of 20 micrograms at time \(t=0\), how much is left after 2000 years?

Find the center and the radius of the circle given by the equation $$ (x-2)^{2}+y^{2}=16 $$

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=3 x^{2} $$

Find the center and the radius of the circle given by the equation $$ (x+1)^{2}+(y-3)^{2}=9 $$

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=x^{6} $$

After 7 days, a particular radioactive substance decays to half of its original amount. Find the decay rate of this substance.

Find the center and the radius of the circle given by the equation $$ 0=x^{2}+y^{2}-4 x+2 y-11 $$

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=5 x^{3} $$

After 5 days, a particular radioactive substance decays to \(37 \%\) of its original amount. Find the half-life of this substance.

Find the center and the radius of the circle given by the equation $$ x^{2}+y^{2}+2 x-4 y+1=0 $$

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=x^{-2} $$

Polonium \(210\left(\mathrm{Po}^{210}\right)\) has a half-life of 140 days. (a) If a sample of \(\mathrm{Po}^{210}\) has a mass of 300 micrograms, find \(\mathrm{a}\) formula for the mass after \(t\) days. (b) How long would it take this sample to decay to \(20 \%\) of its original amount?

(a) Convert \(75^{\circ}\) to radian measure. (b) Convert \(\frac{17}{17} \pi\) to degree measure.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=6 x^{-1} $$

The half-life of \(C^{14}\) is 5730 years. Suppose that wood found at an archeological excavation site contains about \(35 \%\) as much \(\mathrm{C}^{14}\) (in relation to \(\mathrm{C}^{12}\) ) as does living plant material. Determine when the wood was cut.

(a) Convert \(-15^{\circ}\) to radian measure. (b) Convert \(\frac{3}{4} \pi\) to degree measure.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=4 x^{-3} $$

The half-life of \(\mathrm{C}^{14}\) is 5730 years. Suppose that wood found at an archeological excavation site is 15,000 years old. How much \(\mathrm{C}^{14}\) (based on \(\mathrm{C}^{12}\) content) does the wood contain relative to living plant material?

Evaluate the following expressions without using a calculator: (a) \(\sin \left(-\frac{5 \pi}{4}\right)\) (b) \(\cos \left(\frac{5 \pi}{6}\right)\) (c) \(\tan \left(\frac{\pi}{3}\right)\)

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=7 x^{-5} $$

The age of rocks of volcanic origin can be estimated with isotopes of argon \(40\left(\mathrm{Ar}^{40}\right)\) and potassium \(40\left(\mathrm{~K}^{40}\right) . \mathrm{K}^{40}\) decays into \(\mathrm{Ar}^{40}\) over time. If a mineral that contains potassium is buried under the right circumstances, argon forms and is trapped. Since argon is driven off when the mineral is heated to very high temperatures, rocks of volcanic origin do not contain argon when they are formed. The amount of argon found in such rocks can therefore be used to determine the age of the rock. Assume that a sample of volcanic rock contains \(0.00047 \% \mathrm{~K}^{40}\). The sample also contains \(0.000079 \% \mathrm{Ar}^{40}\). How old is the rock? (The decay rate of \(\mathrm{K}^{40}\) to \(\mathrm{Ar}^{40}\) is \(\left.5.335 \times 10^{-10} / \mathrm{yr} .\right)\)

Evaluate the following expressions without using a calculator: (a) \(\sin \left(\frac{3 \pi}{4}\right)\) (b) \(\cos \left(-\frac{13 \pi}{6}\right)\) (c) \(\tan \left(\frac{4 \pi}{3}\right)\)

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ f(x)=3 x^{1} $$

(Adapted from Moss, 1980) \(\quad\) Hall (1964) investigated the change in population size of the zooplankton species Daphnia galeata mendota in Base Line Lake, Michigan. The population size \(N(t)\) at time \(t\) was modeled by the equation $$ N(t)=N_{0} e^{r t} $$ where \(N_{0}\) denotes the population size at time \(0 .\) The constant \(r\) is called the intrinsic rate of growth. (a) Plot \(N(t)\) as a function of \(t\) if \(N_{0}=100\) and \(r=2\). Compare your graph against the graph of \(N(t)\) when \(N_{0}=100\) and \(r=3\). Which population grows faster? (b) The constant \(r\) is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length 1, and you find that at the beginning there were 200 individuals and after one unit of time there were 250 individuals. Determine \(r\). [Hint: Consider the ratio \(N(t+1) / N(t) .]\)

(a) Find the values of \(\alpha \in[0,2 \pi)\) that satisfy $$ \sin \alpha=-\frac{1}{2} \sqrt{3} $$ (b) Find the values of \(\alpha \in[0,2 \pi)\) that satisfy $$ \tan \alpha=\sqrt{3} $$

Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function $$ L(x)=L_{\infty}\left(1-e^{-k x}\right) $$ for \(x \geq 0\), where \(L(x)\) is the length of the fish at age \(x\) and \(k\) and \(L_{\infty}\) are positive constants. (a) Use a graphing calculator to graph \(L(x)\) for \(L_{\infty}=20\), for (i) \(k=1\) and (ii) \(k=0.1\). (b) For \(k=1\), find \(x\) so that the length is \(90 \%\) of \(L_{\infty} .\) Repeat for \(99 \%\) of \(L_{\infty} .\) Can the fish ever attain length \(L_{\infty}\) ? Interpret the meaning of \(L_{\infty}\) (c) Compare the graphs obtained in (a). Which growth curve reaches \(90 \%\) of \(L_{\infty}\) faster? Can you explain what happens to the curve of \(L(x)\) when you vary \(k\left(\right.\) for fixed \(\left.L_{\infty}\right)\) ?

(a) Find the values of \(\alpha \in[0,2 \pi)\) that satisfy $$ \cos \alpha=-\frac{1}{2} \sqrt{2} $$ (b) Find the values of \(\alpha \in[0,2 \pi)\) that satisfy $$ \sec \alpha=2 $$

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ N(t)=130 \times 2^{1.2 t} $$

Which of the following functions is one to one (use the horizontal line test)? (a) \(f(x)=x^{2}, x \geq 0\) (b) \(f(x)=x^{2}, x \in \mathbf{R}\) (c) \(f(x)=\frac{1}{x}, x>0\) (d) \(f(x)=e^{x}, x \in \mathbf{R}\) (e) \(f(x)=\frac{1}{x^{2}}, x \neq 0\) (f) \(f(x)=\frac{1}{x^{2}}, x>0\)

Show that the identity $$ 1+\tan ^{2} \theta=\sec ^{2} \theta $$ follows from $$ \sin ^{2} \theta+\cos ^{2} \theta=1 $$

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ I(u)=4.8 u^{-0.89} $$

Show that the identity $$ 1+\cot ^{2} \theta=\csc ^{2} \theta $$ follows from $$ \sin ^{2} \theta+\cos ^{2} \theta=1 $$

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ R(t)=3.6 t^{1.2} $$

(a) Show that \(f(x)=x^{2}+1, x \geq 0\), is one to one, and find its inverse together with its domain. (b) Graph \(f(x)\) and \(f^{-1}(x)\) in one coordinate system, together with the line \(y=x\), and convince yourself that the graph of \(f^{-1}(x)\) can be obtained by reflecting the graph of \(f(x)\) about the line \(y=x\)

Solve \(2 \cos \theta \sin \theta=\sin \theta\) on \([0,2 \pi)\)

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ L(c)=1.7 \times 10^{2.3 c} $$