Selected Answers to Problem Set 5

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1. i) |c - 0.5| < 0.25 implies that both c - 0.5 < 0.25 and -(c - 0.5) < 0.25.
      Then c < 0.75 and c > 0.25.  So the probability that 0.25 < c < 0.75 is 50%.
   
2. v) You can either solve this for r (or s) and find the integral, or convert this
      into polar coordinates.  The former is an integral between 0 and sqrt(2)/2 of sqrt(1/2 - x2).
      Converting to polar coordinates, we find the integral is:

      Or simply realize that the area of a circle is r2*Pi.  In this case,
      r = sqrt(2)/2, so this yields Pi/2.  One-fourth of the circle (in the first quadrant)
      yields Pi/8.

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2. We'll take the real number line between 0 and 1 to represent arriving between 2:00 and 3:00.

We let P(T < t) = |s - r| < t, where s, r are the arrival times.  The plot is as shown:

This is an example with t = 1/2. The area outside the boxes is (1-t)², so the area inside is 1-(1-t)², or 2t - t². Our equation is the derivative of this; we let f(t) = 2 - 2t. The expected value is the integral from 0 to 1 of t times f(t) (divided by the area of the square, which naturally is one).
This answer comes out to 1/3, which in our terms is twenty minutes.

Now, to find the standard deviation, we take the difference of E(t) (=1/3) and t, square it, and multiply by f(t), as we normally do.

The answer we get for the variance is 1/18, so the standard deviation is the square root of 1/18, or 0.2357 (about 14 minutes).

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3. Rewriting the equation y = a + bx in matrix form, we get:

To solve for X, we need to multiply both sides by the adjoint of A. We already proved in class that the adjoint of A is the transpose of A (A^* = A^T). Therefore, A^TAX = A^TY.


We have 4a - b = 8 and -a + 9b = -11. This is a simple linear equation that equates to a = 61/35 and b = -36/35.

Click here for a maple script (pdf format) on calculating the best fit line for this problem.

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4. The same method applies as in problem one. We rewrite the equation y = c + a*sin(Pi*t/6) + b*cos(Pi*t/6) as a matrix equation: AX = Y.

To solve for X, we need to multiply both sides by A^T. Therefore, A^TAX = A^TY.


We have 3a = sqrt(3), 3b = 0.8, and 6c = 5.6. Therefore a = sqrt(3) / 3, b = 4/15, and c = 14/15.

Click here for a maple script (pdf format) on calculating the ocean tide.

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5. First rewrite the original equation to fit a linear system.

On one side of the equation we want p multipled by something and e multiplied by something. Rewriting to fit this idea gives us this:

We can solve just as above then.



Therefore, we get p = 1.454 and e = 0.694.

Click here for a maple script (pdf format) on calculating the orbit of Tentax.

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6.

a)

b)

c)

d)

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7. Here are the four equations plotting x, y, log x, and log y together. Notice that the last one, plotting log x versus log y, is the most linear.

From the linear squares equations we get that in y = ax + b, a = 1.5063 and b = -1.6591. This equates to log(T) = 1.5063 log(r) - 1.6591. Solving for T, we get T = 0.02 r^(1.5063). Recall that these measurements are not exact; we therefore can assume that the exponent on r is 3/2, which equates to T^2 = r^3, Kepler's Law.

Click here for a maple script (pdf format) on calculating Kepler's law.

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8. To find the weighted averages, we should add the number of arrivals together from both airlines:

	Alaska Airline		America West
Destination	% on time	Total Flights	% on time	Total Flights
Los Angeles	88.9	1370	85.6	1370
Phoenix	94.8	5488	92.1	5488
San Diego	91.4	680	85.5	680
San Francisco	83.1	1054	71.3	1054
Seattle	85.8	2408	76.7	2408
Total	90.8	11000	85.5	11000

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