# Add Math SBA

An investigation into a quadratic expression used to represent a parabolic edge in designing a flower garden utilizing calculus to determine the maximum area of the lawn

Purpose of the Project

Mr. Jack is an avid gardener and he is considering a new design for his garden. He has a rectangular lawn measuring 5 metres by 3 metres and wants to dig up part of it to include a flower bed. He desires to have a parabolic edge for the flower bed as shown below in Figure 1.

FIGURE 1

Objectives of Project

To find an expression that would best represent the parabolic edge. To use Integration to find an expression for the area of the lawn. To Differentiate the expression for the area to determine the maximum area of the lawn. To investigate possible lawn design options with the area found in objective

Mathematical Formulation

y

p

h q x FIGURE 2

1) Finding an expression to represent the parabolic edge of the flower bed which is also the parabolic edge of the lawn. Let y= a(x-h)2 +p be the equation of the parabola where p is the maximum value of y…. (FIGURE 2) Since (q,0) lies on the curve (i.e is a true root

(where q is the positive root of the curve)

Substitute into

Which represents the parabolic edge.

2) Finding an expression for the area of the lawn.

Area under the curve =

Area = = [ + ]

= [( =

=

=

Therefore Area of lawn A=

3) Finding an expression for the maximum area of the lawn. Maximum area occurs when =0 A=

= = 0

=-1

= ----=- = -

For the maximum area of the lawn

4) Alternative designs to be considered for the flowerbed while maintaining the maximum area of the lawn.

Other designs can include different shapes such as rectangles , triangles and circles. Area of rectangle =

Area of Triangle =

Area of Circle =

From the expression for the parabolic edge is

And from The area of the lawn is

From The maximum area of the lawn occurs when

Since the plot is 5m long and 3m wide the maximum area for the lawn would occur when and we will try different values for which will be 2.5 ,3 and 4

y-intercept (substituting ) in...

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