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GaryK [48]
2 years ago
11

a farmer has 100m for fence avaibiable, with which he intends to build a pen for his sheep. he intends to create a rectangular p

en against a permanent stone wall. i)show that a=1/2x(100-x) ii) find dA/dx and d2A/dx^2 iii)find the value of x that makes the area as large as possible, and explain how you know that this is a maximum.
Mathematics
1 answer:
Reika [66]2 years ago
7 0

The perimeter of the area of the pen the farmer intends to build for his

ship includes the length of the permanent stone wall.

Response:

i) The length and width of the rectangular pen are; <em>x</em>, and \dfrac{100 - x}{2}, therefore;

  • The area is; A = \dfrac{1}{2} \cdot x \cdot (100 - x)

ii) \hspace{0.5 cm}\dfrac{dA}{dx}  = 50 - x

\dfrac{d^2A}{dx^2} =  -1

iii) The value of <em>x</em> that makes the area as large as possible is x = 50

<h3>How is the function for the area and the maximum area obtained?</h3>

Given:

The length of fencing the farmer has = 100 m

Part of the area of the pen is a permanent stone wall.

Let <em>x</em> represent the length of the stone wall, we have;

2 × Width = 100 m - x

Therefore;

Width, <em>w</em>, of the rectangular pen, w = \mathbf{\dfrac{100 - x}{2}}

Area of a rectangle = Length × Width

Area of the rectangular pen, is therefore;

  • A = x \times \dfrac{100 - x}{2} = \underline{\dfrac{1}{2} \cdot x \cdot (100 - x)}

ii) \hspace{0.5 cm} \mathbf{\dfrac{dA}{dx}}, and \mathbf{\dfrac{d^2A}{dx^2} } are found as follows;

\dfrac{dA}{dx} = \mathbf{\dfrac{d}{dx} \left(  \dfrac{1}{2} \cdot x \cdot (100 - x) \right)} = \underline{50 - x}

\dfrac{d^2A}{dx^2} = \mathbf{ \dfrac{d}{dx} \left( 50 - x\right)} = \underline{-1}

iii) The value of <em>x</em> that makes the area as large as possible is given as follows;

Given that the second derivative, \dfrac{d^2A}{dx^2} =-1, is negative, we have;

At the maximum area, \dfrac{dA}{dx} = \mathbf{0}, which gives;

\dfrac{dA}{dx} = 50 - x = 0

x = 50

  • The value of x that makes the area as large as possible is <em>x</em>  =<u> 50</u>

Learn more about the maximum value of a function here:

brainly.com/question/19021959

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