CS 221 Fall 2011 Problem Set 5

Note: There is no penalty for turning in this problem set late, but submissions will not be accepted after solutions are posted on or about Friday, 9 December.

Please read the submission instructions carefully. Failure to submit the proper items may cost you points.

For any problem you use MATLAB to solve, create a record of your calculations in MATLAB, using the diary command. Name the diary file "PS5-#", where "#" is the problem number, e.g. "1a".

Problem 1: Solving Nonlinear Equations

1. The distance d traveled by a projectile launched at velocity v at angle θ (neglecting air resistance and assuming flat ground) is given by d = (v²/g) sin 2θ, where g is the gravitational constant (use 9.8 meters/second²). Find the angle θ (in degrees) at which a projectile, launched with initial velocity of 100 m/sec, should be launched so that it travels exactly 900 meters.

2. Find the fifth root of 200, i.e. the number x such that x⁵=200.

3. The van der Waals equation relates pressure P (in atm.), volume V (in L), and temperature T (in K) for real gas:

   P = (nRT)/(V-b) - (n²a)/V²

   where n is the number of moles, R = 0.08206 (L atm)/(mole K) is the gas constant, and a and b are material constants. Consider 1.5 moles of Nitrogen (a=1.39 L² atm/mole², b=0.03919 L/mole) at 25° C stored in a pressure vessel. Determine the volume of the vessel if the pressure is 13.5 atm.

Problem 2: Solving Systems of Linear Equations

Solve the following systems of linear equations Ax=b using MATLAB or Excel:

1. A = 
       10  -1   3   0   77
        0   2  -5   3    6
       21  -4   0 -14.5 17
       3   12  1.5  37  -9
       -4   5  -3   22   0
   b = 
       -58
        37
       249
       448
        67
   

2. A = 
     -17.9    -1.9   -12.0   -20.5    -2.9     9.6    -5.8     7.9     8.6     -0.6
       8.4   -21.3    29.0    -3.5     0.2     5.2    -2.9   -13.3   -13.6     -1.9
      -8.8    -8.3     8.2    -8.2    -2.6    -0.2    -8.4   -23.2     4.5     -2.1
       1.0    13.5    13.7   -15.7   -17.5    -0.3   -11.2   -14.4    -8.4     -3.0
      -5.4   -10.7   -10.5     5.0    -2.8    -7.9    25.2     3.3    -3.3      0.2
       3.0     9.6    -4.6     2.8    -8.3    10.1    16.5     3.9     5.5      0.5
      -6.0     1.2    -2.7     0.3    -9.7    -1.3     3.0     4.5    10.3      8.2
       4.8    14.3    10.9   -13.3   -11.5    -7.1   -12.5    -1.3   -11.1     15.2
       7.3   -19.6    -2.7    11.2    -5.3    13.5    -8.6     1.8    12.6      4.6
      17.1    -1.9     7.0     3.5   -20.0    -2.2    -1.7    -4.7     6.6     -2.0
   b = 
      -2.6
    -197.1
    -272.7
    -382.2
      75.4
     239.9
     164.9
    -133.0
     172.9
     -75.0
   

   (Note: For the second problem, a text file containing A is here, and one for b is here.

Problem 3: Curve-Fitting

1. The resistance, R of a tungsten wire as a function of temperature can be modeled with the equation

   R = R₀[1 + α(T-T₀)]

   where R₀ is the resistance corresponding to temperature T₀, and α is the temperature coefficient of resistance. Determine R₀ and α such that the equation will best fit the following data. Use T₀=20° C.

   T (° C)	20	100	180	260	340	420	500
   R (Ω)	500	676	870	1060	1205	1410	1565

2. A hot-wire anemometer is a device for measuring flow velocity by measuring the cooling effect of the flow on the resistance of a hot wire. The data obtained in calibration tests are as follows (also contained in this file). The first column is the velocity u (in ft/sec) and the second column the voltage V in volts.

   4.72    7.18
   12.49   7.30
   20.03   7.37
   28.33   7.42
   37.47   7.47
   41.43   7.50
   48.38   7.53
   55.06   7.55
   66.77   7.58
   59.16   7.56
   54.45   7.55
   47.21   7.53
   42.75   7.51
   32.71   7.47
   25.43   7.44
   8.18    7.28
   

   Determine the coefficients A and B of the exponential function u=Ae^BV that best fit the data. (Hint: transform the equation as suggested by the above table and use MATLAB's polyfit() function.)

3. The following measurements were recorded in a study on the growth of trees.

   Age (years)	5	10	15	20	25	30	35
   Height (m)	5.2	7.8	9	10	10.6	10.9	11.2

   We want to use the data to derive an equation Height = H(Age) to predict tree height as a function of age. Determine which of the nonlinear equations in the above table gives the best fit for this data, and determine the appropriate coefficients.

Submission Instructions

• Create a single document containing the answers to all the problems. (plain text or PDF or Microsoft Word are all OK formats).
• For any problem you use MATLAB to solve, include any scripts you wrote and a record of your calculations in MATLAB, recorded with the diary command. Name the diary file "PS5-#", where "#" is the problem number, e.g. "1a".
• If you use Excel, submit the spreadsheet you used to solve the problem.