Question

A farmer makes two adjacent rectangular paddocks with 200 feet of fencing. a) Express the total area () in square feet as a function of the length () in feet of the shared side

b) Find the maximum total area that can be enclosed.

Answer 1

A = (100 - y)y

Maximum area = 2500 sq. feet.

Explanation:

Let the length of the combined rectangular area is y feet and the shared width is x feet.

So, perimeter of the two rectangle together is (3x + 2y) = 200 {Given}

⇒ 2y = 200 - 3x

⇒
`y= (1)/(2) (200 - 3x)` ...... (1).

a) Now, area of the total plot in sq. feet is
`A = xy = (1)/(2) (200 - 3x)x` ........ (2)

So, this is the expression for area A in terms of length of shared side x.

b) For area to be maximum the condition is
`(dA)/(dx) = 0`

Now, differentiating equation (2) on both sides with respect to x we get

`(dA)/(dx) = 100 - 3x = 0`

⇒ x = 33.33 feet.

So, from equation (1) we get
`y= (1)/(2) (200 - 3x)`

⇒ x = 50 feet.

So, the value of maximum area = 50 × 33.33 = 1666.5 sq. feet. (Answer)