Solve the following system of equations algebraically:
�
2?? − 3?? + 4?? = 20
?? + 4?? − 3?? = −19
4?? + ?? + 2?? = 4
Problem 2: (15 marks)
A company that operates two wineries needs to produce at least 8000, 14000 and
5000 bottles of red, white and rose wine, respectively. Each day, the first winery produces
2000 bottles of red, 3000 bottles of white and 1000 bottles of rose wine, whereas the
second winery produces 1000 bottles each of red and rose and 2000 bottles of white wine.
Suppose it costs $25000 per day to operate the first winery and $20000 per day to operate
the second winery.
a. Assuming the company would like to minimise costs, express the objective function
of this problem. (2 marks)
b. Express the constraints to which the above function is subject as a system of
inequalities. (4 marks)
c. Graph the system of inequalities and find the corner points of the feasible region.
(6 marks)
d. Determine how many days each winery should operate. (3 marks)
Problem 3: (12 marks)
To make resource and evacuation decisions during bushfires, fire managers use the following
equation to estimate the head fire rate of spread (?? measured in km/hour) in uphill grassland:
?? = 11.324 × ??
60 − ?? + ??
× 2(0.1×??)
where ?? measures wind speed (km/hour), ?? measures temperature (degrees Celsius), ??
measures relative humidity (percent) and ?? measures slope angle (in degrees).
a. If the temperature is 30℃, the relative humidity is 20% and the wind speed is 25 km/h,
how fast would a grass fire spread up a hill with an 8° slope? (2 marks)
b. Given the environmental conditions provided in part (a) above, how fast would the
grass fire spread on flat ground? (2 marks)
c. If a grass fire spread up a 12° hill at a rate of 13.5 km/h, when the relative humidity
was 10% and the wind speed was 20 km/h, determine the temperature at the time.
(4 marks)
d. If a grass fire spread uphill at a rate of 15 km/h, when the relative humidity was 28%, wind
speed was 18 km/h and temperature was 36℃, determine the slope of the hill.
(4 marks)
Semester 201
Problem 4: (12 marks)
A muscle has the ability to shorten when a load, such as weight, is imposed on it. Suppose an
equation for muscle contraction is given as:
?? = (?? + ??)(?? + ??)
where ?? is the load imposed on the muscle, ?? is the velocity of the shortening of the muscle
fibres and ??, ??, and ?? are positive constants.
a. Express ?? as a function of ??. (3 marks)
b. Sketch the function you derived in part (a) above, showing all parameters of interest.
(6 marks)
c. Is the rate of change in ?? as a function ?? constant? Briefly explain. (3 marks)
Problem 5: (14 marks)
Consider the following equations, where ?? denotes quantity in thousands of units, ?? denotes
price in dollars and ?? denotes total costs in dollars:
?? = 350 − 50??
?? = 240+70q
a. Express total revenue (??????) as a function of ??. (1 mark)
b. Express total cost (??????) as a function of ??. (1 mark)
c. Express profit (??) as a function of ??. (2 marks)
d. Determine the break-even level(s) of production. (6 marks)
e. Determine the profit maximising level of production. Find the price and profit at this
output. (4 marks)
Problem 6: (10 marks)
Suppose the spread of disease is modelled by ?? = ??(??−1)??
, where ?? represents the
number of presenting cases, ?? is a constant representing the average number of secondary
cases caused by an infectious person and ?? represents time in weeks.
a. If the value of ?? is approximately 3 for smallpox, estimate the number of infected
individuals after two weeks and six weeks. (2 marks)
b. If the value of ?? is approximately 16 for measles, estimate the number of infected
individuals after one week and four weeks. (2 marks)
c. Due to the process of herd immunity, the threshold level of vaccination (??) necessary
to eradicate disease is given by ?? = 1 − 1
??
. Use this equation to explain why
vaccination has led to the eradication of smallpox but not measles. (2 marks)
d. Suppose the value of ?? for COVID-19 is approximately 2.5. How long would it take for
twenty thousand cases to present?