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1.4 Scaling Factor and its Applications

Similar Figures and Shapes

Two shapes are similar if they share the same shape but not the size.

  • Their corresponding angles are equal;
  • Their corresponding sides are in the same ratio.

Calculating the Scaling Factor

  • The scaling factor is a numerical value representing the scale of one shape/object to a similar shape (with a different scale).
  • The scaling factor can be calculated as the proportional difference between the lengths of the two shapes/objects, the square root of the proportional difference between their areas, or the cubic root of the proportional difference between their volumes:

k=\frac{L_{A}}{L_{B}}=\sqrt{\frac{A_{A}}{A_{B}}}=\sqrt[3]{\frac{V_{A}}{V_{B}}}

Note: this calculates the scaling factor for shape/object A with respect to shape/object B. For example, a scaling factor of 2 means A has twice the length of B.

  • The scaling factor is dimensionless (i.e. has no units).
  • k>0The scaling factor is larger than 0.
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1.3 Surface Area and Volume of Composite Shapes

Note: if you cannot remember the formulas for the surface area and volume of common shapes, revise notes for 1.1 Surface Area of Common Shapes and 1.2 Volume of Common Shapes.

Guide to Finding the Surface Area of a Composite Shape

  • Composite shapes are shapes that are created by merging multiple simple shapes.
  • Due to the large amount of combinations possible, it is not possible to derive formulas for each.
  • To find the surface area of a composite shape, you need to first identify the simple shapes it is made up of. The surface area of the composite shape is equal to the sum of each shape, minus the surface area which has been removed to merge it into the shape:
Read More »1.3 Surface Area and Volume of Composite Shapes

6.2 Methods for Optimum Assignment

Note: if you cannot remember how to represent matching problems in graphs and matrices, revise notes for 6.1 Introduction to Matching Problems.

Optimum Assignment by Inspection

  • In smaller scale matching problems, it may be possible to determine the optimum assignment by inspection.
  • One method is to find the lowest weight assignment for each task, object, etc. being assigned to. If there is any conflict, then consider other tasks, objects, etc.

Example

Teacher

Banjo

Violin

Piano

Dave

10

25

35

Jill

11

15

40

Steve

10

18

46

 

Above is a matrix representing the time in days required for each teacher at a school to teach a beginner’s course in certain instruments. Only one teacher is required for each course and each teacher must be assigned to a course. As this is a smaller-scale problem, we will use inspection to find the optimum assignment.

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6.1 Introduction to Matching Problems

Matching Problems

  • There are many instances in which a several tasks, objects, or other needs to be assigned to a group of people, objects or other. For example, individually assigning a group of people a list of tasks. This is known as a matching problem.
  • Matching problems often have multiple solutions however there is generally one or more optimum solution. This is the one which achieves the intended result while expending the least time or resources.
  • In some cases, multiple assignments may be required for either group (e.g. a task may require many people or a person may be assigned to many tasks).

Bipartite Graph

Read More »6.1 Introduction to Matching Problems

5.2 Introduction to Dijkstra’s Algorithm

Dijkstra’s Algorithm

  • Dijkstra’s algorithm provides a methodical method for determining the shortest path between two vertices.
  • Dijkstra’s algorithm works as follows:

1.  Construct a table consisting of a row of empty elements where each column corresponds to each vertex in the network except the starting vertex, and the first (and for now only) row corresponds to the starting vertex.

2.  In each element of the row, write the weight of the edge connecting the starting vertex to the vertex of the corresponding column. If there is no edge such edge for a vertex, place a cross in that element instead.

3.  Draw a square around the lowest weight in the row.

Read More »5.2 Introduction to Dijkstra’s Algorithm

3.3 Guide to Minimal Connector Problems

Note: if you cannot remember how to construct minimum spanning trees using Prim’s algorithm, revise notes for 3.2 Minimum Spanning Trees.

Guide to Analysing Minimal Connector Problems

  • Minimal connector problems are problems which require the use of a minimum spanning tree.
  • Always use Prim’s algorithm unless otherwise stated as inspection is more error prone and generally slower.
  • It is generally easier to select a starting node with a small amount of edges.
Read More »3.3 Guide to Minimal Connector Problems

3.2 Minimum Spanning Trees

Note: if you cannot remember what a spanning tree is, revise notes for 3.1 Trees and Spanning Trees.

Minimum Spanning Trees

  • Most connected graphs will have multiple spanning trees. In a weighted graph, one of these spanning trees will have the lowest weight (i.e. the sum of each edge’s weight is the least). This is known as a minimum spanning tree.
  • Minimum spanning trees are commonly used for optimisation problems. For example, when transferring products between towns, it is advantageous to know the fastest way of doing so.

By Inspection

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2.3 Hamiltonian Paths and Cycles

Note: if you cannot remember the concepts of walks, paths and cycles, revise notes for 2.1 Introduction to Walks.

Hamiltonian Paths

  • A Hamiltonian path is a path which visits every vertex of a graph exactly once.
  • Hamiltonian paths are useful for situations when every vertex must be visited but the route taken does not necessarily matter. For example, when a message must be sent to every individual in a communication network.
  • There are no convenient methods (within the scope of Further Maths) for determining if a Hamiltonian path exists in a network and it must be determined through observation.

Example

A company has 6 distribution centres, named centre A-F. A delivery driver must deliver the company’s product to each centre. To optimise time, the driver plots a Hamiltonian path between the centres: B-A-D-C-E-F, as shown below:

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2.2 Eulerian Trails and Circuits

Note: if you cannot remember the features of walks, trails and circuits, revise notes for 2.1 Introduction to Walks.

Eulerian Trails

  • A Eulerian trail is a trail which encompasses every edge of a graph.
  • A Eulerian trail will exist if the graph:
    • Is connected.
    • Has exactly two vertices with an odd degree.
    • Eulerian trails are useful in situations where every edge must be visited, for example when planning a mail route.
    • A Eulerian trail will start from one of the odd vertices.

Example

Read More »2.2 Eulerian Trails and Circuits

5.3 Modelling Annuity Investments using Technology

Note: if you cannot remember how to model reducing balance relations with regular repayments, revise notes for 5.1 Modelling Annuity Investments.

Guide to Analysing Annuity Investments using Technology (Casio Graphics Calculator)

Note: if you cannot remember how to use the Casio Financial Calculator, revise notes for 3.3 Modelling Reducing Balance Systems with Regular Repayments using Technology.

  • The annual interest rate should be entered/calculated as a positive value.
  • The initial value (PV) should be entered/calculated as a negative value (remember we justify this by saying we must lose this amount to create the investment).
  • The payment (PMT) (i.e. the amount withdrawn) should be entered/calculated as negative (we can justify this by saying we lose this amount per compounding period to the investment)
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