Exam (elaborations)
Mathematical Methods FIA1 Problem Solving Modelling Task
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This assignment was awarded 20/20 for QCE (Queensland) Mathematical Methods, for year 11.
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PSMT 2022.Introduction:
To construct an irrigation system that waters three, evenly spaced rows of crops, a farmer will utilise
a mathematical model to optimise the path of water; the shape can be modulated by adjusting the
angle that water propagates from the origin of the irrigation head.
The purpose of the task will be to investigate and deduce three different equations that will model
the path of water, to ensure all crops are adequately watered.
Formulate:
Initial assumptions and observations:
- Environmental determinants (type of crops, weather, soil type, sun exposure, etc…) are
irrelevant, and will not be considered in mathematical modelling.
- For simplicity, all rows of crops are on level terrain.
- Cost of irrigation heads is irrelevant and will not be considered in mathematical modelling.
- For simplicity, flow/pressure rate of irrigation system will not be considered in mathematical
modelling.
- Parameters to the size of area are considered theoretically undefined.
- For consistency in the task’s local context, units of measurement will be expressed in SI
units.
- Final values deduced will be expressed to three decimal places.
- For practical reasonability, the shape of water adheres to a parabolic curve.
- For simplicity, the path of water will intersect with the base of crop, on the ground.
It is important to assume that the crops are not initially planted; mathematically, it is assumed that
the distance between crops will be explicated by the mathematical model, and not as an initial
variable of the model.
Other contingencies to be considered are water pressure and rate of flow. Although such aspects do
affect the shape of water as variables of velocity, trajectory, and range, especially in this – a scenario
of projectile motion, it is observed that in the context of this task, it is not relevant to consider such.
Once data on the shape of water produced by irrigation heads is acquired by…
- Photographs/images
- Numerical data provided of distance of throw, or maximum height of trajectory.
…it will be tabulated in a table or annotated appropriately, using conversion software and photo
editing applications. It is assumed that the path of water will adhere to the curve of a parabola,
indicating that…
- Quadratic formulas must be deduced must be by means of substitution, and/or
simultaneous equations
- Equations must be expressed in standard form f ( x)=a x2 +bx +c , and in turning point form
f (x)=a ¿,
Provided the nature of equations as mathematical models, such should be expressed in function
notation and graphed using technology, such as Desmos – or otherwise specified, to visualise
, solutions; given the context of equations to model physical phenomena, justified domains and
ranges of (x) and ( y ) values will need to be explicated.
Solve:
It is observed that the angle of water propagation is adjustable; Hunter Industries is a manufacturer
of commercial and residential irrigation products – including adjustable irrigation nozzles for farming
and agriculture.
Their PGJ model of rotary sprinkles has been selected as the irrigation system. As Hunter Industries
is an American subsidiary, all data tabulated from the website was converted from Imperial units to
SI units, using conversion software (see Appendix 1).
Hunter Industries denotes three important data of the PGJ model: the height of the physical model –
the origin of the water; the maximum height of trajectory of water, and the distance it occurs from
the irrigation head – the vertex of the path of water.
A sample of three adjustable nozzle heads with three different initial angles of trajectory is denoted
below…
Table 1: Sample of processed data of maximum height (metres) and distance of throw (metres) of
PGJ models, per degrees (degrees) of trajectory from base of irrigation head (Hunter Industries,
2021).
Model Degrees Of Trajectory Max Height Of Spray Distance from Head to Maximum Height
10 0.61 1.22
PGJ 12 1.52 4.88
15 1.83 7.32
The height of the model is 0.1m, or 10cm. With no particular criteria, the first row of data will be
selected to deduce equations from.
The data elucidates the ( y )-intercept of the graph and the coordinates of the vertex. Hence, an
equation can be deduced in turning point form y=a ( x−h )2 +k .
When x=1.219 , y=0.610
2 [ 1]
Hence , f ( x )=a ( x−1.219 ) + 0.610
When x=0 , y=0.1
Substitute bothvalues into [ 1 ] ¿ deduce ' a ' .
2
∴ 0.1=a ( 0−1.22 ) + 0.61
0.1=(a)(1.48 …)+0.61
−0.51= ( a ) ( 1.48 … )
a=−0.343
2
∴ f (x )=−0.343 ( x−1.219 ) +0.610
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