Afence is to be built to enclose a rectangular area of 260 square feet. the fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 14 dollars per foot. find the dimensions of the enclosure that is most economical to construct.

There are several ways to minimize or maximize a function.

The dimensions of the enclosure that is most economical to construct are 27.1 ft by 9.6 ft

Let the sides of the rectangle be x and y.

Such that:

The area is given as: 260 square feet.

So, we have:

Make x the subject

The perimeter (P) of the rectangle is:

Expand

Split

Three sides cost $3 per foot, while the fourth costs $14 per foot.

So, the cost function is:

Substitute

Rewrite as:

Differentiate

Set to 0

Subtract 20 from both sides

Divide both sides by -1560

Cancel out negatives

Take inverse of both sides

Take the square roots of both sides

Recall that:

Substitute 9.6 for y

Hence, the dimensions of the enclosure that is most economical to construct are 27.1 ft by 9.6 ft

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Length of the rectangle = 9.58 feet and width = 27.14 foot

Step-by-step explanation:

Let the length of the rectangular area is = x feet

and the width of the area = y feet

Area of the rectangle = xy square feet

Or xy = 260

y = -------(1)

Cost to fence the three sides = $3 per foot

Therefore cost to fence one length and two width of the rectangular area

= 3(x + 2y)

Similarly cost to fence the fourth side = $14 per foot

So, the cost of the remaining length = 14x

Total cost to fence = 3(x + 2y) + 14x

Cost (C) = 3(x + 2y) + 14x

C = 3x + 6y + 14x

  = 17x + 6y

From equation (1)

C =

Now we take the derivative,

C' = 17 -

To minimize the cost of fencing,

C' = 0

17 - = 0

= 17

x = 9.58 foot

and y =

y = 27.14 foot

step-by-step explanation:

try remembering the steps

b and c

step-by-step explanation: