Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

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

Sagot :

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; x, and [tex]\dfrac{100 - x}{2}[/tex], therefore;

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

[tex]ii) \hspace{0.5 cm}\dfrac{dA}{dx} = 50 - x[/tex]

[tex]\dfrac{d^2A}{dx^2} = -1[/tex]

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

How is the function for the area and the maximum area obtained?

Given:

The length of fencing the farmer has = 100 m

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

Let x represent the length of the stone wall, we have;

2 × Width = 100 m - x

Therefore;

Width, w, of the rectangular pen, [tex]w = \mathbf{\dfrac{100 - x}{2}}[/tex]

Area of a rectangle = Length × Width

Area of the rectangular pen, is therefore;

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

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

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

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

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

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

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

[tex]\dfrac{dA}{dx} = 50 - x = 0[/tex]

x = 50

  • The value of x that makes the area as large as possible is x  = 50

Learn more about the maximum value of a function here:

https://brainly.com/question/19021959

We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.