Question

A fence 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.

Answer 1

Answer:

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 = $\frac{260}{x}$ -------(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 = $17x+\frac{260\times 6}{x}$

Now we take the derivative,

C' = 17 - $\frac{1560}{x^{2} }$

To minimize the cost of fencing,

C' = 0

17 - $\frac{1560}{x^{2} }$ = 0

$\frac{1560}{x^{2} }$ = 17

$x^{2} =\frac{1560}{17}$

$x=\sqrt{\frac{1560}{17} }$

x = 9.58 foot

and y = $\frac{260}{9.58}$

y = 27.14 foot