3.3.2
Modelling with Parametric Equations
Modelling with Parametric Equations
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 and 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 seconds is modelled by the parametric equations:
- Show that the motion of the remote-controlled boat is a straight line.
- Find the speed of the remote-controlled boat.
Rearrange equation
Rearrange equation
- To find the Cartesian equation, rearrange the equation for to get
on its own:
Find the Cartesian equation
Find the Cartesian equation
- Substitute this into the equation for gives:
Inspection
Inspection
- The equation is a straight line with gradient 3 and a -intercept of 0.
- and 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 =
Find the speed in each direction
Find the speed in each direction
- In the -direction, the speed is:
- Speed = m/s
- In the -direction, the speed is:
- Speed = 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 = m/s
- Remember to include units in your answers!
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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