Chapter 1

In each of the questions 54-56, use a measured power law to make predictions. Estimating the mass of extinct animals is very difficult, since estimates have to be made based on fossil skeletons and without knowledge of the muscles and other tissues that make up much of an animal's mass. van Valkenburgh (1990) measured body mass and skull length in living carnivores and found that, if \(y\) is body mass measured in \(\mathrm{kg}\) and \(x\) is skull length measured in \(\mathrm{mm}\), $$ y=k x^{3.13} \text { . } $$ (a) A gray wolf ( Canis lupus) weighs \(45 \mathrm{~kg}\), and has a skull length of \(275 \mathrm{~mm} .\) Calculate the coefficient of proportionality \(k\) in the power law. (b) Estimate the weight of the sabretooth tiger (Smilodon populator \() .\) S. populator fossils have skull lengths of \(350 \mathrm{~mm}\). (c) How strong was Smilodon's bite? Hartstone-Rose, Perry, \mathrm{\\{} a n d ~ M o r r o w ~ ( 2 0 1 2 ) ~ s t u d i e d ~ t h e ~ c o r r e l a t i o n ~ b e t w e e n ~ b i t e ~ f o r c e ~ and body mass. They found that, if bite force (measured in newtons, or \(\mathrm{N}\) ) is \(f\) and body mass (measured in \(\mathrm{kg}\) ) is \(y\), then $$ f=c y^{0.96} $$ for some constant \(c\). (i) The tiger has a bite force of \(7980 \mathrm{~N}\), and a mass of \(200 \mathrm{~kg}\). Calculate the constant \(c\). (ii) Estimate the bite force of Smilodon populator using your body mass estimate from part (b).

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

Gomes et al. (1999) found a power law relationship between land area and language diversity. Specifically, if \(A\) is the area of a region measured in \(\mathrm{km}^{2}\) and \(D\) is the number of different languages spoken by the people occupying that region. then $$ D=0.2 A^{0.41} . $$ (a) Estimate from this formula the number of languages spoken in the United States (area: \(9.857 \times 10^{6} \mathrm{~km}^{2}\) ). [In reality, there are 337 languages within the United States, but your answer from (a) agrees well with the 176 indigenous languages spoken within the United States.] (b) Equation (1.8) is a statistical relationship rather than a strict mathematical formula. To see this, make a plot showing both equation (1.8) and the following data points on the same axes. $$ \begin{array}{lrc} \hline \text { Country } & \text { Area (km }^{2} \text { ) } & \text { # Languages } \\ \hline \text { Cameroon } & 475,400 & 230 \\ \text { Belgium } & 30,500 & 3 \\ \text { China } & 9,597,000 & 129 \\ \text { India } & 3,287,600 & 122 \\ \text { Mexico } & 1,943,900 & 60 \\ \text { Kenya } & 569,100 & 68 \\ \text { United Arab Emirates } & 83,600 & 7 \\ \hline \end{array} $$ (i) How many countries lie near the line given by the formula? (ii) Are there any countries that do not lie on the line given by the formula? (iii) Use equation (1.8) to predict the number of languages in the UCLA campus (the area of the UCLA campus is \(\left.1.7 \mathrm{~km}^{2}\right)\). Does your answer make sense?

Find the equation of a circle with center \((1,-2)\) and radius 2 .

When log \(y\) is graphed as a function of log.x, a straight line results. Graph straight lines, each given by two points, on a log-log plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(2,7),\left(x_{2}, y_{2}\right)=(7,2) $$

Reis et al. (2010) investigated how the frequency of tongue motion \((f\), number of licks per second \()\) for a cat drinking water depends on the cat's mass \((M\), measured in \(\mathrm{kg})\). They found that $$ f=c M^{-1 / 6} $$ that is, frequency of licks is proportional to (body mass) \(^{-1 / 6}\). (a) Given the following data $$ \begin{array}{lcc} \hline \text { Species } & \text { Mass (kg) } & \text { Frequency }\left(\mathbf{s}^{-1}\right) \\ \hline \text { Domestic cat } & 4.8 & 3.55 \\ \text { Tiger } & 175.0 & 1.70 \\ \hline \end{array} $$ calculate the constant \(c\) in (1.9). (b) Calculate the lapping frequency for Professor R.'s slightly tubby pet cat, Texas Ranger (mass, \(M=6.4 \mathrm{~kg}\) ). (c) Calculate the lapping frequency for a cheetah (mass, \(M=\) \(40 \mathrm{~kg})\). (d) Estimate from equation (1.9) the mass of a cat whose lapping frequency is less than \(1 \mathrm{~s}^{-1}\).

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, on a log-log plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(4,2),\left(x_{2}, y_{2}\right)=(8,8) $$

Assume that a population size at time \(t\) is \(N(t)\) and that \(N(t)=2^{t}, t \geq 0\) (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 \(4 .\) (b) Where does the circle intersect the \(y\) -axis? (c) Does the circle intersect the \(x\) -axis? Explain.

Assume that a population size at time \(t\) is \(N(t)\) and that \(N(t)=20 \cdot 2^{t}, t \geq 0\). (a) Find the population size at time \(t=0\). (b) Show that \(N(t)=20 e^{t \ln 2}, t \geq 0\) (c) How long will it take until the population size reaches \(1000 ?\) \([\) Hint \(:\) Find \(t\) so that \(N(t)=1000 .]\)

(a) Find all possible radii of a circle centered at \((2,-5)\) so that the circle intersects only one axis. (b) Find all possible radii of a circle centered at \((2,-5)\) so that the circle intersects both axes.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=2 x^{5} $$

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 \((x+2)^{2}+y^{2}=25\).

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=3 x^{2} $$

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 \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=x^{6} $$

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 \(0=x^{2}+y^{2}+6 x+2 y-12\). (To do this, you must complete the squares.)

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=2 x^{-2} $$

After 4 days, a particular radioactive substance decays to \(30 \%\) of its original amount. Find the half-life of this substance.

Find the center and the radius of the circle given by the equation \(x^{2}+y^{2}+2 x-4 y+1=0\). (To do this, you must complete the squares.)

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=2 x^{-2} $$

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 100 micrograms, find a formula for the mass after \(t\) days. (b) How long would it take this sample to decay to \(10 \%\) of its original amount? (c) Sketch the graph of the amount of mass left after \(t\) days.

(a) Convert \(65^{\circ}\) to radian measure. (b) Convert \(\frac{11 \pi}{12}\) to degree measure.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=6 x^{-1} $$

The half-life of \(\mathrm{C}^{14}\) 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 \(\frac{7}{4} \pi\) to degree measure.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=4 x^{-3} $$

The half-life of \(\mathrm{C}^{14}\) is 5730 years. Suppose that wood found at an archeological excavation site is 10,000 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) \(\sin \left(-\frac{\pi}{4}\right)\) (b) \(\cos \left(\frac{5 \pi}{6}\right)\) (c) \(\tan \left(\frac{\pi}{6}\right)\)

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=9 x^{-3} $$

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 \(5.335 \times 10^{-10} / \mathrm{yr}\).)

Evaluate the following expressions without using a calculator: (a) \(\sin \left(\frac{5 \pi}{4}\right)\) (b) \(\cos \left(-\frac{11 \pi}{6}\right)\) (c) \(\tan \left(\frac{\pi}{3}\right)\)

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. $$ f(x)=3 x^{1} $$

A growing population contains \(N(t)\) individuals \(N(t)\) at time \(t\) is 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) You are given the following data for the size of the population. $$ \begin{array}{cc} \hline \boldsymbol{t} & \boldsymbol{N}(\boldsymbol{t}) \\ \hline 0 & 100 \\ 2 & 300 \\ \hline \end{array} $$ (i) Calculate the parameters \(N_{0}\) and \(r\) to make the mathematical model fit the data. (ii) When will the population size first reach 1000 individuals? (iii) When will the population size first reach 10,000 individuals?

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

Rundle et al. (2003) showed that earthquakes in Southern California obey an exponential distribution-that is, if \(N(m)\) is the number of earthquakes in a given year whose magnitude exceeds \(m\), then $$ N(m)=c \cdot 10^{-m} $$ where \(c\) is a positive constant. (a) Suppose in a given year there are 10 earthquakes of magnitude 5 or above. (i) Calculate the constant \(c\). (ii) How many earthquakes will have magnitudes exceeding 2 ? ( 2 is the threshold at which earthquakes can be felt by most people.) (iii) How many earthquakes will have magnitude exceeding 6 ? (6 is the threshold for an earthquake to be regarded as strong.)

(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 \(\sec \alpha=\sqrt{2}\).

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) \(f(x)=\sqrt{x}, x \geq 0\) (d) \(f(x)=\ln x, x>0\) (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. $$ I(u)=4.8 u^{-0.89} $$

(a) Show that \(f(x)=x^{3}-1, x \in \mathbf{R}\), 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 \(y=x\).

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 \(\cos ^{2} \theta-2=2 \sin \theta\) 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. $$ L(t)=2^{-2.7 y+1} $$