Question

You work as a marine engineer, your job is to design several types of fuels for marine crafts. You do this
by combining three different chemicals in different proportions; chemical x, chemical y, and chemical z.
Your chemicals are stored in vats of equal volume (10 litres).
You are ordering a shipment of new chemicals, and need to work out the density (mass per volume)
of each chemical to calculate freight costs. You look through the documentation that you have on the
chemicals. Although you don’t find a list of their respective densities, you do find a ledger where you
have recorded the number of vats of each chemical required to produced test batches of three different
types of fuel (150 litres of each), and the mass of the fuel produced (in kilograms):
No. vats of chemical x No. vats of chemical y No. vats of chemical z Fuel mass (kg), 150L

	No. vats of chemical x	No. vats of chemical y	No. vats of chemical z	Fuel mass (kg), 150L
Fuel A	8	1	6	80
Fuel B	3	5	7	91
Fuel C	4	9	2	84

(a) By forming an appropriate matrix system and applying Gaussian elimination to an augmented matrix,
show that the mass per vat of chemical x, chemical y and chemical z is 4, 6 and 7 kilograms respectively.
Express your working in fractions to avoid rounding error during to Gaussian elimination.
(b) Assuming that no mass or volume is lost when producing the fuel, determine the densities of each
chemical.
(c) You need to make 150,000 litres of each fuel type. You need to choose between Freight Company
A and Freight Company B; Freight Company A charges $0.08 per kilogram of chemical delivered for
chemicals up to 0.5 kg/L dense, and $0.09 per kilogram for chemicals more dense than 0.5 kg/L whereas
Freight Company B charges a flat rate of $0.05 per litre of chemical delivered.
i) Calculate the total freight cost and use this information to identify which freight company will
be cheaper given that chemicals can not be mixed prior to freighting.
ii) Suppose that you still need to produce 150,000 litres of Fuel A and Fuel C, but now want
to produce more than 150,000 litres of Fuel B. How many litres of Fuel B could you produce before
switching freight companies becomes cheaper?

Answer 1