Question

A fence is to be built to enclose a rectangular area of 210 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 12 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Answer 1

Answer:

Length of enclosure =18.89 foot

Width of enclosure=11.12 foot

Step-by-step explanation:

We are given that a fence is to be built to enclose a rectangular area 210 square feet.

Fence along three sides is to be made of material that costs 5 dollars per foot

and the material for the fourth side costs 12 dollars per foot.

We have to find the dimension of the enclosure that is most economical

Let x be the length and y be the width  of enclosure

We know that area of rectangle=$x\timesy$

$xy=210$

$y=\frac{210}{x}$

Cost of four sides =$2(5x)+5(y)+12(y)$

Total cost=$10x+17y$

C=$10x+17\cdot\frac{210}{x}$

C=$10x+\frac{3570}{x}$

Differentiate w.r.t x

$\frac{dC}{dt}=10-\frac{3570}{x^2}$

Substitute$\frac{dC}{dx}=0$

$10-\frac{3570}{x^2}=0$

$\frac{3570}{x^2}=10$

$x^2=357$

$x=\sqrt{357}$

x=18.89

Differentiate w.r.t x

$\frac{d^2C}{dx^2}=\frac{7140}{x^3}$

Substitute x=18.89

Then we get

$\frac{d^2C}{dx^2}=\frac{7140}{(18.89)^3} >0$

Hence, the cost is minimum.

Length of enclosure =18.89 foot

Width of enclosure=$\frac{210}{18.89}=11.12 foot$

Hence, the dimension of the enclosure that is most economical to construct

Length=18.89 foot and width=11.12 foot

Answer 2

Answer with explanation:

Let the length of rectangular fence = x meters

Let the width of the rectangular fence = y meters

Thus

$Area=x\times y=210ft^{2}...........(i)$

Now Let the total cost to construct the fence be 'C' can be obtained if we construct the shorter side with material with cost $12 per foot

$\therefore C=5\times (2y+x)+12\times x\\\\C=10y+17x...............(ii)$

Using value of x from the equation i into equation ii we get

$C=10y+17\times \frac{210}{y}$

hence to minimize the cost we differentiate both sides with respect to 'y' and equate the result to zero thus we get

$C=10y+17\times \frac{210}{y}\\\\\frac{dC}{dy}=\frac{d}{dy}(10y+\frac{3570}{y})\\\\0=10-\frac{3570}{y^{2}}\\\\\therefore y=\sqrt{\frac{3570}{10}}=18.894feet$

Thus $x=\frac{210}{18.894}=11.11feet$

Thus value of length = 11.114 feet and value of width = 18.894 feet.