This assignment aims to develop a mathematical model for the placement of lines on a fishing rod

Authors Avatar by avsinglai (student)

MATHS ASSIGNMENT

FISHING ROD – MODELLING TASK

This assignment aims to develop a mathematical model for the placement of lines on a fishing rod by investigating different methods (matrix methods, polynomial functions, and technology) that model a given set of data and discovering the equation that best models the data.

Leo has a fishing rod with length 230cm.

The given data about his rod is:

Guide number: 1, 2, 3, 4, 5, 6, 8

Distance from tip (cm): 10, 23, 38, 55, 74, 96, 120, 149

Define suitable variables, discuss parameters / constraints.

x (independent variable): Guide number from the tip of the rod (1, 2, 3, 4, 5, 6, 7, 8)

y (dependent variable): Distance from the tip in centimeters (10, 23, 38, 55, 74, 96, 120, 149)

Constraints on x: Whole number; Positive number; Greater than or equal to 1; Smaller than or equal to 8  (may not need this last requirement cause we may add guide?)

Constraints on y: Real number; Positive number; Less than length of fishing rod (230cm); Space accommodated for reel and handle further limits the space between the guides

Since x and y cannot be negative, the plotted graph will be limited to the first quadrant.

Using technology, plot the data points on a graph.

(insert graph)

The scatter plot resembles a curve and probably represents a part of a quadratic function.

Let f(x) = y.

Notice that:

f(2) – f(1) = 15

f(3) – f(2) = 17

f(4) – f(3) = 19

And the difference between these are all 2.

Since after 2 steps of calculating the differences a repeating answer is reached, f(x) is probably quadratic.

See: http://www.purplemath.com/modules/nextnumb.htm


Using matrix methods or otherwise, find a quadratic function which models this situation.

Explain the process you used.

On a new set of axes draw these model functions and the original data points.

Comment on any differences.

Quadratic function form: y=ax²+bx+c

We need to find a, b and c using matrix methods in order to find the quadratic function.

Since there are 3 variables, we can create a 3 by 3 (3 x 3) matrix and a 3 by 1 (3 x 1) matrix.

Answer 1 (simplest)

Choose 3 data points from the given table above.

(1, 10), (2, 23), (3, 38)

Form 3 equations:

a(1)²+b(1)+c=10

a(2)²+b(2)+c=23

a(3)²+b(3)+c=38

which is

a + b + c = 10

4a + 2b + c = 23

9a + 3b + c = 38

Transform the equations into matrices:

1…1…1…|10

4…2…1…|23

9…3…1…|38

Take -4 x row 1 and add to row 2 to get a new row 2

Take -9 x row 1 and add to row 3 to get a new row 3

1…1…1…|10

0..-2..-3…|-17

0..-6..-8…|-52

Divide row 2 by -2 to get a new row 2

1…1…1…|10

0…1…3/2…|17/2

0...-6...-8…|-52

Add – row 2 to row 1 to get a new row 1

Add 6 x row 2 to row 3 to get a new row 3

1…0...-1/2…|3/2

0…1...3/2…|17/2

0…0…1…|-1

Take 1/2 x row 3 add to row 1 to get a new row 1

Take -3/2 x row 3 add to row 2 to get a new row 2

1…0…0…|1

0…1…0…|10

0…0…1…|-1

a = 1, b = 10 and c = -1

Substitute into y = ax² + bx + c

y = x² + 10x -1

Therefore the quadratic function is y=x²+10x-1.

Join now!

(insert graph with model function and original data points)

The graph goes through the first 5 data points and gets close to the last 3 data points only.

Answer 2

Choose 3 data points from the given table above.

To make sure the function would pass through the first and last points and to increase the accuracy of the function, choose the first and last set of data, together with another set of data in between the two, that represent the spread of the original data points.

(1, 10), (5, 74), (8, 149)

(similar process as ...

This is a preview of the whole essay