Mathematics

 

Modelling with Quadratics

Many real life situations or events can be modelled pretty accurately by mathematics formulae.

For instance, the tidal height of the River Thames that flows through London Suburbs has been modelled by a simple variation of the sine cure, which takes the following shape:

Thames tidal height modelling

In the same way, the motion of a projectile can easily be modelled by a quadratic relationship.

In the following animation, a stone, thrown down from a cliff, is modelled by a quadratic equation - height against time. You can change the time and see how the height changes with it.
h(t) = 12 - t2/8, where h(t) is the height when time is t.

Modelling with Quadratics - worked examples

E.g.1

The height of a ball, dropped from a tree top, is modelled by the equation, h(t) = 625 - t2/25. Find the following:

  1. The significance of the number 625
  2. Time taken by the ball to drop by 4 metres
  3. Time taken by the ball to hit the ground
  1. h(t) = 625 - t2/25
    h(o) = 625
    It's the height of the tree.
  2. h(t) = 625 - t2/25
    621 = 625 - t2/25
    4 = t2/25
    t2 = 100
    t = ± 10
    Since t is positive, t = 10 seconds.
  3. h(t) = 625 - t2/25
    h(t) = 0
    625 - t2/25 = 0
    t2 = 625 x 25
    t = ± 125 seconds.
    Since t is positive, t = 125 seconds.

E.g.2

An arrow was thrown from the top of a hill; its height can be modelled by, h(t) = -4t2 + 16t + 80. Find the following:

  1. The height of the hill
  2. Time taken by the arrow to reach maximum height
  3. Time taken by the arrow to be at a height of 90 from the hill
  4. Time taken by the arrow to hit the ground
  1. h(t) = -4t2 + 16t + 80
    h(o) = 80
    It's the height of the hill.
  2. h(t) = -4t2 + 16t + 80 = -4[t2 - 4t - 20] = -4[(t - 2)2 - 24] = 4(t - 2)2 + 96
    By the completing the square, when the height it maximum, t = 2. The corresponding height = 96 metres.
  3. h(t) = -4t2 + 16t + 80
    h(t) = 90
    -4t2 + 16t + 80 = 90
    -4t2 + 16t - 10 = 0
    t = 0.8 seconds or t = 3.2 seconds.
  4. h(t) = -4t2 + 16t + 80
    -4t2 + 16t + 80 = 0
    t = -2.8 or t = 6.8
    t = 6.8 seconds.

 

 

Recommended Reading


 

Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7th edition in print.

Recommended - GCSE & iGCSE


 

This is the best book available for the new GCSE(9-1) specification and iGCSE: there are plenty of worked examples; a really good collection of problems for practising; every single topic is adequately covered; the topics are organized in a logical order.

Recommended for A Level


 

This is the best book that can be recommended for the new A Level - Edexcel board: it covers every single topic in detail;lots of worked examples; ample problems for practising; beautifully and clearly presented.