# Simon has 160 meters of fencing to build a rectangular garden The garden's area (in square meters) as

###### Question:

## Answers

40 metres

Step-by-step explanation:

Given the garden's area (in square meters) as a function of the garden's width x (in meters) modeled by the equation A(x)=−x(x−80), the width that will produce the maximum garden area will occue at dA(x)/dx = 0

Given A(x)=−x(x−80)

A(x) = -x²+80x

dA(x)/dx = -2x+80

-2x+80 = 0

-2x = -80

Dividing both sides by -2

-2x/-2 = -80/-2

x = 40 metres

Hence the width that will produce the maximum garden area is 40 metres

the maximum area is 1600 square meters

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Step-by-step explanation:

It is given in the question , that Simon has 160 meters of fencing to build a rectangular garden. the garden's area (in square meters) as a function of the garden's width w (in meters) is modeled by

[tex]A(w) = -w(w-80) \\ A(w) = -w^2 + 80w[/tex]

Which represents parabola and the parabola is maximum at its vertex, that is

[tex]w = -\frac{b}{2a} = - \frac{80}{2} = 40 meters[/tex]

Therefore the width is 40 meters and the area is

[tex]A(40) = -40(40-80) = -40*-40 = 1600 meter ^2[/tex]

Simon has 160 meters of fencing to build a rectangular garden. the garden's area (in square meters) as a function of the garden's width www (in meters) is modeled by a(w)=-w(w-80) what is the maximum area possible?

Solution:

Let width of the rectangular garden=w

Perimeter of the rectangular garden=2(length+width)

160=2(length+w)

Divide by 2 on both sides

80=length+w

So, Length= 80-w

So, Area of rectangular garden= Length* Width

Area, A(w)=(80-w)(w)

Area, A(w)=-w²+ 80 w

Area is a quadratic equation. And, quadratic equation makes a parabola.

For the maximum area, We need to find the vertex of the parabola.

The formula for x-coordinate of the vertex=[tex]\frac{-b}{2a}[/tex]

x-coordinate of vertex=[tex]\frac{-(80)}{2(-1)}[/tex]

x-coordinate of vertex=40

So, Width= 40 meters

Length=80-w=80-40=40 meters

Maximum area possible=Length* Width=40*40=1600 square meters