Math 2250 Lab 5
Name/Unid:
Due Date: 2 October 2014
1. (Fitting curves to data)
(a) There is a parabola with the equation y = ax2 + bx + c that passes through the
points (1, 6), (−2, −6) and (3, 4). Exhibit and solve the system of three equations in
the three unknowns a, b, c. Use Gaussian elimination to compute the reduced row
echelon form of the augmented matrix, in order to find a, b, c.
(b) If there is another parabola that passes throught the points (1, −6), (−2, 6) and
(3, −4), write down the equation of the parabola. (Hint: use (a).)
(c) Sketch by hand the resulting parabolas in (a) and (b), together with the given
points they pass through.
2. The balancing of chemical reactions always leads to systems of linear equations. The
equations are often ”sparse” enough that the method of substitution will work well, but
Gaussian elimination is guaranteed to work.
(a) The complete combustion of propene C3 H6 with oxygen O2 results in carbon dioxide
CO2 and water H2 O, in such a way that the total numbers of each element are the
same before and after the reaction:
a(C3 H6 ) + b(O2 ) → c(CO2 ) + d(H2 O).
The coefficients a, b, c, d for the numbers of each molecule need to be found from the
three equations equating the number of C, H, O atoms on each side. For example,
equating the number of oxygen atoms yields the equation
2b = 2c + d ⇒ 2b − 2c − d = 0.
Find the other two equations, and then use Gaussian elimination to find the solutions to this homogeneous system of three equations in four unknowns. Pick the
free parameter in the solution so that a, b, c, d are each positive integers having
no common factors. This will give you the usual balancing equation for complete
combustion of propene.
(b) If there is not enough oxygen for the amount of propene, combustion will be incomplete and the intermediate product carbon monoxide CO will be created but
not completely converted into carbon dioxide CO2 . This is potentially dangerous
in home heating. The more complicated partial combustion equation is
a(C3 H6 ) + b(O2 ) → c(CO2 ) + d(H2 O) + e(CO).
If we set a = 1 this equation becomes
(C3 H6 ) + b(O2 ) → c(CO2 ) + d(H2 O) + e(CO).
Solve for c, d, e in terms of the parameter b. Note that all four unknowns must
be non-negative, so that b can only be in a certain interval. Discuss the range of
outcomes as b varies in this interval.
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3. (a) A commercial customer has ordered 108 gallons of paint that contains an equal
amount of red paint, yellow paint, green paint, and blue paint, and hence could be
prepared by mixing 27 gallons of each color. However, the store wished to prepare
this order by mixing four types of paint that are already available in large quantity:
Paint A that is a mixture of 30.5% red, 15% yellow, 20.5% green, and 34% blue
paint;
Paint B that is 26% red, 28.5% yellow, 40% green, and 5.5% blue paint;
Paint C that is 5% red, 60% yellow, 15% green, and 20% blue; and
Paint D that is 21.5% red, 25.5% yellow, 18% green, and 35% blue.
How many gallons of paints A, B, C, D are needed to prepare the customers order?
Write out a system of equations that represents the information given, then solve
that system using Maple or Matlab.
(b) Now there is another customer. The new customer wants to order 108 gallons of
paint that cotains 70% of red paints, 10% of each of yellow, green and blue paints.
The store again wants to prepare the paint for the customer from Paints A, B, C
and D. Is it possible for the store to do this time? Explain your answer by solving
a system of linear equations with Maple or Matlab.
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