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Farmer wants to build a rectangular corral next to a straight river with the river serving as one side of the corral. If the farmer has only 40 40 m of fencing with which to build the corral, what dimensions will maximize its area? Express the area A ( w ) A ( w ) of the corral as a function of its width w w .

Answer :

Answer:

Dimensions of the rectangular corral will be 20m × 10m.

Step-by-step explanation:

Farmer wants to build a rectangular corral next to a straight river.

So, to build a coral farmer will use the fence on three sides only (excluding river side).

Let length of the rectangular corral = l m

and width of the corral = w m

Area of the corral A = lw  ------(1)

Since length of the fence that the farmer has = 40 m

So, 40 = l + w + w

l + 2w = 40

l = 40 - 2w -----(2)

Now we substitute the value of w in equation (1)

[tex]A=w(40 - 2w)[/tex]

We can write the area in the form of the function as

[tex]A(w)=w(40-2w)[/tex]

For maximum area we will find the derivative of the area and equate it to zero.

[tex]A'(w)=40-4w[/tex]

A'(w) = 0

40 - 4w = 0

w = 10 m

From equation (2),

l = 40 - 2(10)

l = 20 m

Therefore, dimensions of the rectangular corral will be 20m × 10m

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