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Section 4.7 Projects for Chapter 4

Project 4.3 Bode's Law

In 1772, the astronomer Johann Bode promoted a formula for the orbital radii of the six planets known at the time. This formula calculated the orbital radius, \(r\text{,}\) as a function of the planet's position, \(n\text{,}\) in line from the Sun. (Source: Bolton, 1974)

  1. Evaluate Bode's law, \(r (n) = 0.4 + 0.3(2^{n-1})\text{,}\) for the values in the table. (Use a large negative number, such as \(n = -100\text{,}\) to approximate \(r (-\infty)\text{.}\))

    \(n\) \(-\infty\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\)
    \(r(n)\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
  2. How do the values of \(r(n)\) compare with the actual orbital radii of the planets shown in the table? (The radii are given in astronomical units (AU). One AU is the distance from the Earth to the Sun, about \(149.6\times 10^6\) kilometers.) Assign values of \(n\) to each of the planets so that they conform to Bode's law.

    Planet Mercury Venus Earth Mars Jupiter Saturn
    Orbitan radius (AU) \(0.39\) \(0.72\) \(1.00\) \(1.52\) \(5.20\) \(9.54\)
    \(n\) \(\) \(\) \(\) \(\) \(\) \(\)
  3. In 1781, William Herschel discovered the planet Uranus at a distance of \(19.18\) AU from the Sun. If \(n = 7\) for Uranus, what does Bode's law predict for the orbital radius of Uranus?

  4. None of the planets’ orbital radii corresponds to \(n = 2\) in Bode's law. However, in 1801 the first of a group of asteroids between the orbits of Mars and Jupiter was discovered. The asteroids have orbital radii between \(2.5\) and \(3.0\) AU. If we consider the asteroids as one planet, what orbital radius does Bode's law predict?

  5. In 1846, Neptune was discovered \(30.6\) AU from the Sun, and Pluto was discovered in 1930 \(39.4\) AU from the Sun. What orbital radii does Bode's law predict for these planets?

Project 4.4 Plague

In 1665, there was an outbreak of the plague in London. The table shows the number of people who died of plague during each week of the summer that year. (Source: Bolton, 1974)

Week Deaths Week Deaths
\(0\text{,}\) May 9 \(9\) \(12\text{,}\) August 1 \(2010\)
\(1\text{,}\) May 16 \(3\) \(13\text{,}\) August 8 \(2817\)
\(2\text{,}\) May 23 \(14\) \(14\text{,}\) August 15 \(3880\)
\(3\text{,}\) May 30 \(17\) \(15\text{,}\) August 22 \(4237\)
\(4\text{,}\) June 6 \(43\) \(16\text{,}\) August 29 \(6102\)
\(5\text{,}\) June 13 \(112\) \(17\text{,}\) September 5 \(6988\)
\(6\text{,}\) June 20 \(168\) \(18\text{,}\) September 12 \(6544\)
\(7\text{,}\) June 27 \(267\) \(19\text{,}\) September 19 \(7165\)
\(8\text{,}\) July 4 \(470\) \(20\text{,}\) September 26 \(5533\)
\(9\text{,}\) July 11 \(725\) \(21\text{,}\) October 3 \(4929\)
\(10\text{,}\) July 18 \(1089\) \(22\text{,}\) October 10 \(4327\)
\(11\text{,}\) July 25 \(1843\) \(\)
  1. Scale horizontal and vertical axes for the entire data set, but plot only the data for the first 8 weeks of the epidemic, from May 9 through July 4. On the same axes, graph the function \(f (x) = 2.18 (1.83)^x\text{.}\)

  2. By what weekly percent rate did the number of victims increase during the first eight weeks?

  3. Add data points for July 11 through October 10 to your graph. Describe the progress of the epidemic relative to the function \(f\) and offer an explanation.

  4. Make a table showing the total number of plague victims at the end of each week and plot the data. Describe the graph.

Project 4.5 Koch snowflake

The Koch snowflake is an example of a fractal. It is named in honor of the Swiss mathematician Niels Fabian Helge von Koch (1870–1924). Here is how to construct a Koch snowflake:

  • Draw an equilateral triangle with sides of length \(1\) unit. This is stage \(n = 0\text{.}\)
  • Divide each side into \(3\) equal segments and draw a smaller equilateral triangle on each middle segment, as shown in the figure. The new figure (stage \(n = 1\)) is a \(6\)-pointed star with \(12\) sides.
  • Repeat the process to obtain stage \(n = 2\text{:}\) Trisect each of the \(12\) sides and draw an equilateral triangle on each middle third.
  • If you continue this process forever, the resulting figure is the Koch snowflake.
Koch snowflake
  1. We will consider several functions related to the Koch snowflake:

    \(S(n)\) is the length of each side in stage \(n\)
    \(N(n)\) is the number of sides in stage \(n\)
    \(P(n)\) is the perimeter of the snowflake at stage \(n\)

    Fill in the table describing the snowflake at each stage.

    Stage \(n\) \(S(n)\) \(N(n)\) \(P(n)\)
    \(0\)
    \(1\)
    \(2\)
    \(3\)
  2. Write an expression for \(S(n)\text{.}\)

  3. Write an expression for \(N(n)\text{.}\)

  4. Write an expression for \(P(n)\text{.}\)

  5. What happens to the perimeter as \(n\) gets larger and larger?

  6. As \(n\) increases, the area of the snowflake increases also. Is the area of the completed Koch snowflake finite or infinite?

Project 4.6 Sierpinski carpet

The Sierpinski carpet is another fractal. It is named for the Polish mathematician Waclaw Sierpinski (1882–1969). Here is how to build a Sierpinski carpet:

  • Start with a unit square (sides of length 1 unit.)
  • For stage \(n = 1\text{,}\) trisect each side and partition the square into 9 smaller squares. Remove the center square, leaving a hole surrounded by 8 squares, as shown in the figure.
  • For stage \(n = 2\text{,}\) repeat the process on each of the remaining 8 squares.
  • If you continue this process forever, the resulting is the Sierpinski carpet.
Sierpinski carpet
  1. We will consider several functions related to the Sierpinski carpet:

    \(S(n)\) is the side of a new square at stage \(n\)
    \(A(n)\) is the area of a new square at stage \(n\)
    \(N(n)\) is the number of new squares removed at stage \(n\)
    \(R(n)\) is the total area removed at stage \(n\)
    \(T(n)\) is the total area remaining at stage \(n\)

    Fill in the table describing the carpet at each stage.

    Stage \(n\) \(S(n)\) \(A(n)\) \(N(n)\) \(R(n)\) \(T(n)\)
    \(0\)
    \(1\)
    \(2\)
    \(3\)
  2. Write an expression for \(S(n)\text{.}\)

  3. Write an expression for \(A(n)\text{.}\)

  4. Write an expression for \(N(n)\text{.}\)

  5. Write an expression for \(R(n)\text{.}\)

  6. Write an expression for \(T(n)\text{.}\)

  7. What happens to the area remaining as \(n\) approaches infinity?

Project 4.7 Stream order

The order of a stream or river is a measure of its relative size. A first-order stream is the smallest, one that has no tributaries. Second-order streams have only first-order streams as tributaries. Third-order streams may have first- and second-order streams as tributaries, and so on. The Mississippi River is an example of a tenth-order stream, and the Columbia River is ninth order.

Both the number of streams of a given order and their average length are exponential functions of their order. In this problem, we consider all streams in the United States. (Source: Leopold, Luna, Gordon, and Miller, 1992)

  1. Using the given values, find a function \(N(x) = ab^{x-1}\) for the number of streams of a given order.

  2. Complete the column for number of streams of each order. (Round to the nearest whole number of streams for each order.)

  3. Find a function \(L(x) = ab^{x-1}\) for the average lengthof streams of a given order, then complete that column.

  4. Find the total length of all streams of each order, hence estimating the total length of all stream channels in the United States.

Order Number Average Length Total Length
\(1\) \(1,600,000\) \(1\)
\(2\) \(339,200\) \(2.3\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(8\)
\(9\)
\(10\)
Project 4.8 Species rank

Related species living in the same area often evolve in different sizes to minimize competition for food and habitat. Here are the masses of eight species of fruit pigeon found in New Guinea, ranked from smallest to largest. (Source: Burton, 1998)

Size rank \(1\) \(2\) \(3\) \(4\)
Mass (grams) \(49\) \(76\) \(123\) \(163\)
Size rank \(5\) \(6\) \(7\) \(8\)
Mass (grams) \(245\) \(414\) \(592\) \(802\)
  1. Plot the masses of the pigeons against their order of increasing size. What kind of function might fit the data?

  2. Compute the ratios of the masses of successive sizes of fruit pigeons. Are the ratios approximately constant? What does this information tell you about your answer to part (a)?

  3. Compute the average ratio to two decimal places. Using this ratio, estimate the mass of a hypothetical fruit pigeon of size rank \(0\text{.}\)

  4. Using your answers to part (c), write an exponential function that approximates the data. Graph this function on top of the data and evaluate the fit.

In Projects 7 and 8, we will prove the formulas in Section 4.5 for the present and future values of an annuity.

The future value of \(\$M\) of money is its value in the future: its current value plus the interest it will accrue in the interval.

The present value of \(\$M\) of money is the amount you would need to deposit now so that it will grow to \(\$M\) in the future.

Project 4.9 Future value

Suppose you deposit \(\$100\) at the end of every \(6\) months into an account that pays \(4\%\) compounded annually. How much money will be in the account at the end of \(3\) years?

  1. During the \(3\) years, you will make \(6\) deposits. Use the formula \(F = P(1 + \dfrac{r}{n})^{nt}\) to write an expression for the future value (principal plus interest) of each deposit. (Do not evaluate the expression!)

    Deposit
    number
    Amount
    deposited
    Time in
    account
    Future
    value
    \(1\) \(100\) \(2.5\) \(100(1.02)^5\)
    \(2\) \(100\) \(2\) \(\)
    \(3\) \(100\) \(1.5\) \(\)
    \(4\) \(100\) \(1\) \(\)
    \(5\) \(100\) \(0.5\) \(\)
    \(6\) \(100\) \(0\) \(\)
  2. Let \(S\) stand for the sum of the future values of all the deposits. Write out the sum, without evaluating the terms you found in part (a).

    \begin{equation*} S = \hphantom{000000} \end{equation*}
  3. You could find \(S\) by working out all the terms and adding them up, but what if there were \(100\) terms, or more? We will use a trick to find the sum in an easier way. Multiply both sides of the equation in part (b) by \(1.02\text{.}\) (Use the distributive law on the right side!)

    \begin{equation*} 1.02S = \hphantom{000000} \end{equation*}
  4. Now subtract the equation in part (b) from the equation in part (c). Be sure to line up like terms on the right side.

    \begin{align*} 1.02S \amp=\amp\amp~~~~\\ -S \amp=\\ 0.02S \amp= \end{align*}
  5. Finally, solve for \(S\text{.}\) If you factor \(100\) from the numerator on the right side, your expression should look a lot like the formula for the future value of an annuity. (To help you see this, note that, for this example, \(\dfrac{r}{n}=\text{?} \) and \(nt=\text{?} \))

  6. Try to repeat the argument above, using letters for the parameters instead of numerical values.

Project 4.10 Present value

You would like to set up an account that pays \(4\%\) interest compounded semiannually so that you can withdraw \(\$100\) at the end of every \(6\) months for the next \(3\) years. How much should you deposit now?

  1. During the \(3\) years, you will make \(6\) withdrawals. Use the formula \(P = A(1 + \dfrac{r}{n})^{-nt}\) to write an expression for the present value of those withdrawals. (Do not evaluate the expression!)

    Withdrawal
    number
    Amount
    withdrawn
    Time in
    account
    Present
    value
    \(1\) \(100\) \(0.5\) \(100(1.02)^{-1}\)
    \(2\) \(100\) \(1\) \(\)
    \(3\) \(100\) \(1.5\) \(\)
    \(4\) \(100\) \(2\) \(\)
    \(5\) \(100\) \(2.5\) \(\)
    \(6\) \(100\) \(3\) \(\)
  2. Let \(S\) stand for the sum of the present values of all the withdrawals. Write out the sum, without evaluating the terms you found in part (a).

    \begin{equation*} S = \hphantom{000000} \end{equation*}
  3. We will use a trick to evaluate the sum. Multiply both sides of the equation in part (b) by \(1.02\text{.}\) (Use the distributive law on the right side!)

    \begin{equation*} 1.02S = \hphantom{000000} \end{equation*}
  4. Now subtract the equation in part (b) from the equation in part (c). Be sure to line up like terms on the right side.

    \begin{align*} 1.02S \amp=\amp\amp~~~~\\ -S \amp=\\ 0.02S \amp= \end{align*}
  5. Finally, solve for \(S\text{.}\) If you factor \(100\) from the numerator on the right side, your expression should look a lot like the formula for the present value of an annuity. (To help you see this, note that, for this example, \(\dfrac{r}{n}=\text{?} \) and \(nt=\text{?} \))

  6. Try to repeat the argument above, using letters for the parameters instead of numerical values.