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Modelling with Parametric Equations

Maths: Edexcel A Level Pure Maths, topic 3.3.2

Modelling with Parametric Equations

Parametric equations are used to model real-life situations.

Models

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.
Time

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.
Example

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.
Rearrange $$x$$ equation

Rearrange xx equation

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

Find the Cartesian equation

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

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.
Find the speed of the boat

Find the speed of the boat

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

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
Find the magnitude of the speed

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!

Test yourself on Modelling with Parametric Equations

  • The design of a skateboard ramp is given by the parametric equations:
    x=3cos⁡t,  y=3sin⁡t+3,  −π2≤t≤0x = 3\cos t,\;y=3\sin t+3, \;-\frac{\pi}{2}\leq t\leq0
    Where xx and yy are the horizontal and vertical length in metres respectively.

    What is the maximum height of the ramp?

  • An artist designs a sculpture using the parametric equations:
    x=13t+cos⁡t,  y=7sin⁡t,  0≤t≤πx = 13t +\cos t,\;y=7\sin t, \;0\leq t\leq\pi
    What is the height of the sculpture when t=π6t = \frac{\pi}{6}?

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