3.3.2

Modelling with Parametric Equations

Test yourself

Modelling with Parametric Equations

Parametric equations are used to model real-life situations.

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Models

  • A mathematical model is a set of equations that describe a real-life situation or object.
  • Answering a question about a model that uses parametric equations will require you to give your answer in units that are relevant to the question.
  • You may also have to comment on how realistic a model is in the context of the question.
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Time

  • We can define a set of parametric equations with time as the parameter.
  • Then xx and yy are defined as functions of time that describe the two-dimensional motion of an object.
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Example

  • A remote-controlled boat is driven on the surface of a lake.
  • The position (in metres) of the boat at time tt seconds is modelled by the parametric equations:
    • x=5t,      y=15t,      t>0x = 5t,\;\;\;y=15t,\;\;\;t>0
  • Show that the motion of the remote-controlled boat is a straight line.
  • Find the speed of the remote-controlled boat.
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Rearrange xx equation

  • To find the Cartesian equation, rearrange the equation for xx to get tt on its own:
    • x=5tt=x5x = 5t \rightarrow t = \frac{x}{5}
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Find the Cartesian equation

  • Substitute this into the equation for yy gives:
    • y=15÷x5=3xy = 15 \div \frac{x}{5} = 3x
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Inspection

  • The equation y=3xy = 3x is a straight line with gradient 3 and a yy-intercept of 0.
  • xx and yy are the position of the boat, so the boat is moving in a straight line.
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Find the speed of the boat

  • Remember the equation for speed:
    • Speed = distancetime\frac{\text{distance}}{\text{time}}
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Find the speed in each direction

  • In the xx-direction, the speed is:
    • Speed = 5tt=5\frac{5t}{t} = 5 m/s
  • In the yy-direction, the speed is:
    • Speed = 15tt=15 \frac{15t}{t} = 15 m/s
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Find the magnitude of the speed

  • The overall size of the speed of the boat is given by Pythagoras' theorem:
    • Speed = (5)2+(15)2=15.81\sqrt{(5)^2+(15)^2} = 15.81 m/s
  • Remember to include units in your answers!

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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