Chapter 13

Environmental Control The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutant as a result of industrial waste and that \(10 \%\) of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after \(n\) years is given by the recursively defined sequence \(p_{0}=250 \quad p_{n}=0.9 p_{n-1}+15\) Determine the amount of pollutant in the lake after 2 years. That is, determine \(p_{2}\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Solve: }(x+3)^{2}=(x+3)(x-5)+7 $$

If you have been hired at an annual salary of \(\$ 42,000\) and expect to receive annual increases of \(3 \%,\) what will your salary be when you begin your fifth year?

Fibonacci Sequence Let \(u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}\) define the \(n\) th term of a sequence. (a) Show that \(u_{1}=1\) and \(u_{2}=1\). (b) Show that \(u_{n+2}=u_{n+1}+u_{n}\) (c) Draw the conclusion that \(\left\\{u_{n}\right\\}\) is a Fibonacci sequence.

A pendulum swings through an arc of length 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length. (a) What is the length of the arc of the 10 th swing? (b) On which swing is the length of the arc first less than 1 foot? (c) After 15 swings, what total length has the pendulum swung? (d) When it stops, what total length has the pendulum swung?

Approximating \(f(x)=e^{x}\) In calculus, it can be shown that $$ f(x)=e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !} $$ We can approximate the value of \(f(x)=e^{x}\) for any \(x\) using the following sum $$ f(x)=e^{x} \approx \sum_{k=0}^{n} \frac{x^{k}}{k !} $$ for some \(n\). (a) Approximate \(f(1.3)\) with \(n=4\). (b) Approximate \(f(1.3)\) with \(n=7\). (c) Use a calculator to approximate \(f(1.3)\) (d) Using trial and error, along with a graphing utility's SEOuence mode, determine the value of \(n\) required to approximate \(f(1.3)\) correct to eight decimal places.

Don contributes \(\$ 500\) at the end of each quarter to a tax-sheltered annuity (TSA). What will the value of the TSA be after the 80 th deposit ( 20 years) if the per annum rate of return is assumed to be \(5 \%\) compounded quarterly?

Bode's Law In \(1772,\) Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun: $$ a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2} $$ where \(n \geq 2\) is the number of the planet from the sun. (a) Determine the first eight terms of the sequence. (b) At the time of Bode's publication, the known planets were Mercury \((0.39 \mathrm{AU}),\) Venus \((0.72 \mathrm{AU}),\) Earth \((1 \mathrm{AU})\) Mars \((1.52 \mathrm{AU}),\) Jupiter \((5.20 \mathrm{AU}),\) and Saturn \((9.54 \mathrm{AU})\) How do the actual distances compare to the terms of the sequence? (c) The planet Uranus was discovered in \(1781,\) and the asteroid Ceres was discovered in \(1801 .\) The mean orbital distances from the sun to Uranus and Ceres " are \(19.2 \mathrm{AU}\) and \(2.77 \mathrm{AU},\) respectively. How well do these values fit within the sequence? (d) Determine the ninth and tenth terms of Bode's sequence. (e) The planets Neptune and Pluto" were discovered in 1846 and \(1930,\) respectively. Their mean orbital distances from the sun are \(30.07 \mathrm{AU}\) and \(39.44 \mathrm{AU},\) respectively. How do these actual distances compare to the terms of the sequence? (f) On July \(29,2005,\) NASA announced the discovery of a dwarf planet \((n=11),\) which has been named Eris. Use Bode's Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.

Droste Effect The Droste Effect, named after the image on boxes of Droste cocoa powder, refers to an image that contains within it a smaller version of the image, which in turn contains an even smaller version, and so on. If each version of the image is \(\frac{1}{5}\) the height of the previous version, the height of the \(n\) th version is given by \(a_{n}=\frac{1}{5} a_{n-1}\). Suppose a Droste image on a package has a height of 4 inches. How tall would the image be in the 6 th version?

Reflections in a Mirror A highly reflective mirror reflects \(95 \%\) of the light that falls on it. In a light box having walls made of the mirror, the light reflects back-and-forth between the mirrors. (a) If the original intensity of the light is \(I_{0}\) before it falls on a mirror, write the \(n\) th term of the sequence that describes the intensity of the light after \(n\) reflections. (b) How many reflections are needed to reduce the light intensity by at least \(98 \% ?\)

For a child born in 2018 , the cost of a 4 -year college education at a public university is projected to be \(\$ 185,000 .\) Assuming a \(4.75 \%\) per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have \(\$ 185,000\) in 18 years when the child begins college?

Show that $$ 1+2+\cdots+(n-1)+n=\frac{n(n+1)}{2} $$ [Hint: Let $$ \begin{array}{l} S=1+2+\cdots+(n-1)+n \\ S=n+(n-1)+(n-2)+\cdots+1 \end{array} $$Add these equations. Then $$ 2 S=[1+n]+[2+(n-1)]+\cdots+[n+1] $$

In an old fable, a commoner who had saved the king's life was told he could ask the king for any just reward. Being a shrewd man, the commoner said, "A simple wish, sire. Place one grain of wheat on the first square of a chessboard, two grains on the second square, four grains on the third square, continuing until you have filled the board. This is all I seek." Compute the total number of grains needed to do this to see why the request, seemingly simple, could not be granted. (A chessboard consists of \(8 \times 8=64\) squares.

\(\sqrt{8}\)

Suppose that, throughout the U.S. economy, individuals spend \(90 \%\) of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is \(0.90 .\) For example, if Jane earns an additional dollar, she will spend \(0.9(1)=\$ 0.90\) of it. The individual who earns \(\$ 0.90\) (from Jane) will spend \(90 \%\) of it, or \(\$ 0.81 .\) This process of spending continues and results in an infinite geometric series as follows: $$1,0.90,0.90^{2}, 0.90^{3}, 0.90^{4}, \ldots$$ The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend \(90 \%\) of every additional dollar that they earn?

\(\sqrt{21}\)

\(\sqrt{89}\)

One method of pricing a stock is to discount the stream of future dividends of the stock. Suppose that a stock pays \(\$ P\) per year in dividends, and historically, the dividend has been increased \(i \%\) per year. If you desire an annual rate of return of \(r \%,\) this method of pricing a stock states that the price that you should pay is the present value of an infinite stream of payments: $$\text { Price }=P+P \cdot \frac{1+i}{1+r}+P \cdot\left(\frac{1+i}{1+r}\right)^{2}+P\cdot\left(\frac{1+i}{1+r}\right)^{3}+\cdots$$ The price of the stock is the sum of an infinite geometric series. Suppose that a stock pays an annual dividend of \(\$ 4.00\), and historically, the dividend has been increased \(3 \%\) per year. You desire an annual rate of return of \(9 \%\). What is the most you should pay for the stock?

Triangular Numbers A triangular number is a term of the sequence $$ u_{1}=1 \quad u_{n+1}=u_{n}+(n+1) $$ List the first seven triangular numbers.

A special section in the end zone of a football stadium has 2 seats in the first row and 14 rows total. Each successive row has 2 seats more than the row before. In this particular section, the first seat is sold for 1 cent, and each following seat sells for \(5 \%\) more than the previous seat. Find the total revenue generated if every seat in the section is sold. Round only the final answer, and state the final answer in dollars rounded to two decimal places. (JJC) \(^{\dagger}\)

Challenge Problem If the terms of a sequence have the property that \(\frac{a_{1}}{a_{2}}=\frac{a_{2}}{a_{3}}=\cdots=\frac{a_{n-1}}{a_{n}},\) show that \(\frac{a_{1}^{n}}{a_{2}^{n}}=\frac{a_{1}}{a_{n+1}}\)

Suppose \(x, y, z\) are consecutive terms in a geometric sequence. If \(x+y+z=103\) and \(x^{2}+y^{2}+z^{2}=6901,\) find the value of \(y\)

Investigate various applications that lead to a Fibonacci sequence, such as in art, architecture, or financial markets. Write an essay on these applications.

Koch's Snowflake The area inside the fractal known as the Koch snowflake can be described as the sum of the areas of infinitely many equilateral triangles, as pictured below. For all but the center (largest) triangle, a triangle in the Koch snowflake is \(\frac{1}{9}\) the area of the next largest triangle in the fractal. Suppose the area of the largest triangle has area of 2 square meters. (a) Show that the area of the Koch snowflake is given by the series $$A=2+2 \cdot 3\left(\frac{1}{9}\right)+2 \cdot 12\left(\frac{1}{9}\right)^{2}+2 \cdot 48\left(\frac{1}{9}\right)^{3}+2 \cdot 192\left(\frac{1}{9}\right)^{4}+\cdots$$ (b) Find the exact area of the Koch snowflake by finding the sum of the series.

You are interviewing for a job and receive two offers for a five-year contract: A: \(\$ 40,000\) to start, with guaranteed annual increases of \(6 \%\) for the first 5 years B: \(\$ 44,000\) to start, with guaranteed annual increases of \(3 \%\) for the first 5 years Which offer is better if your goal is to be making as much as possible after 5 years? Which is better if your goal is to make as much money as possible over the contract (5 years)?

If \(\$ 2500\) is invested at \(3 \%\) compounded monthly, find the amount that results after a period of 2 years.

Which of the following choices, \(A\) or \(B\), results in more money? A: To receive \(\$ 1000\) on day \(1, \$ 999\) on day \(2, \$ 998\) on day \(3,\) with the process to end after 1000 days B: To receive \(\$ 1\) on day \(1, \$ 2\) on day \(2, \$ 4\) on day 3 , for 19 days

You have just signed a 7-year professional football league contract with a beginning salary of \(\$ 2,000,000\) per year. Management gives you the following options with regard to your salary over the 7 years. 1\. A bonus of \(\$ 100,000\) each year 2\. An annual increase of \(4.5 \%\) per year beginning after 1 year 3\. An annual increase of \(\$ 95,000\) per year beginning after 1 year Which option provides the most money over the 7-year period? Which the least? Which would you choose? Why?

For \(\mathbf{v}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j},\) find the dot product \(\mathbf{v} \cdot \mathbf{w}\).

Suppose you were offered a job in which you would work 8 hours per day for 5 workdays per week for 1 month at hard manual labor. Your pay the first day would be 1 penny. On the second day your pay would be two pennies; the third day 4 pennies. Your pay would double on each successive workday. There are 22 workdays in the month. There will be no sick days. If you miss a day of work, there is no pay or pay increase. How much do you get paid if you work all 22 days? How much do you get paid for the 22nd workday? What risks do you run if you take this job offer? Would you take the job?

Find the horizontal asymptote, if one exists, of \(f(x)=\frac{9 x}{3 x^{2}-2 x-1}\)

In a triangle, angle \(B\) is 4 degrees less than twice the measure of angle \(A,\) and angle \(C\) is 11 degrees less than three times the measure of angle \(B .\) Find the measure of each angle.

Make up two infinite geometric series, one that has a sum and one that does not. Give them to a friend and ask for the sum of each series.

Find the average rate of change of \(y=\tan \left(\sec ^{-1} x\right)\) over the interval \(\left[\frac{\sqrt{10}}{3}, \sqrt{10}\right] .\)

Describe the similarities and differences between geometric sequences and exponential functions.

If \(f(x)=5 x^{2}-2 x+9\) and \(f(a+1)=16,\) find the possible values for \(a\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Use the Change-of-Base Formula and a calculator to evaluate \(\log _{7} 62\). Round the answer to three decimal places.

In calculus, the critical numbers for a function are numbers in the domain of \(f\) where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined. Find the critical numbers for \(f(x)=\frac{x^{2}-3 x+18}{x-2}\) if \(f^{\prime}(x)=\frac{x^{2}-4 x-12}{(x-2)^{2}}\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the unit vector in the same direction as \(\mathbf{v}=8 \mathbf{i}-15 \mathbf{j}\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the value of the determinant: $$\left|\begin{array}{rrr}3 & 1 & 0 \\ 0 & -2 & 6 \\ 4 & -1 & -2\end{array}\right|$$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Liv notices a blue jay in a tree. Initially she must look up 5 degrees from eye level to see the jay, but after moving 6 feet closer she must look up 7 degrees from eye level. How high is the jay in the tree if you add 5.5 feet to account for Liv's height? Round to the nearest tenth.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write the factored form of the polynomial function of smallest degree that touches the \(x\) -axis at \(x=4,\) crosses the \(x\) -axis at \(x=-2\) and \(x=1,\) and has a \(y\) -intercept of 4.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Given \(s(t)=-16 t^{2}+3 t,\) find the difference quotient \(\frac{s(t)-s(1)}{t-1}\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find a rectangular equation of the plane curve with parametric equations \(x(t)=t+5\) and \(y(t)=\sqrt{t}\) for \(t \geq 0\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the function \(g\) whose graph is the graph of \(y=\sqrt{x}\) but is stretched vertically by a factor of 7 and shifted left 5 units.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Factor completely: \(x^{4}-29 x^{2}+100\)