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

Answer:

A = (100 - y)y

Maximum area = 2500 sq. feet.

Step-by-step 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

⇒ $y= \frac{1}{2} (200 - 3x)$ ...... (1).

a) Now, area of the total plot in sq. feet is $A = xy = \frac{1}{2} (200 - 3x)x$ ........ (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 $\frac{dA}{dx} = 0$

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

$\frac{dA}{dx} = 100 - 3x = 0$

⇒ x = 33.33 feet.

So, from equation (1) we get $y= \frac{1}{2} (200 - 3x)$

⇒ x = 50 feet.

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