Question

A fence is to be built to enclose a rectangular area of 1250 square feet. the fence along three sides is to be made of material that costs ​$3 per foot. the material for the fourth side costs ​$9 per foot. find the dimensions of the rectangle that will allow for the most economical fence to be built.

Answer 1

Let x and y be the length and width of the rectangle.

Because the area is 1250 ft², therefore
xy = 1250, or
y = 1250/x            

We know that three sides cost $3 per foot, and the fourth side costs $9 per foot.

Case 1: Three sides are (x,y,x) and the 4-th side is y.
The cost is
C = 3(2x +y) + 9y
   = 6x + 12y
   = 6x + 12(1250/x)
   = 6x + 15000/x
For C to be minimum, C' = 0. That is
6 - 15000/x² = 0
x² = 15000/6
x =  50 ft, y = 1250/50 = 25 ft
C = 6(50) + 15000/(50²) = $600

Case 2: Three sides are (y,x,y) and the 4-th side is x.
The cost is
C = 3(x + 2y) + 9x
    = 12x + 6y
    = 12x + 6(1250/x)
    = 12x + 7500/x
For C to be minimum, C' = 0.
12 - 7500/x² = 0
x² = 7500/12 = 625
x = 25 ft, y = 50ft
C = 12(25) + 7500/25 = $600

Answer: 25 ft by 50 ft

Answer 2

Answer:

The dimensions of the rectangle that will allow for the most economical fence to be built is 50 feet and 25 feet in length and width respectively.

Step-by-step explanation:

Given:

Area enclosed by rectangular fence is, $A =1250 \;\rm square\;\rm feet$.

Cost of fencing three sides is, c = $3 per foot.

Cost of fencing fourth side is, c' = $9 per foot.

Let a and b be the length and width of rectangle respectively.

If length of three sides be a, b, a. Then length of fourth side is b.

Then cost is,

$C=c(2a+b)+c'(b)\\C=3(2a+b)+9b\\C=6a+12b$  ...............................................................(1)

Area is,

$A=ab\\1250 = ab\\b=\dfrac{1250}{a}$

Substitute in equation (1) as,

$C=6a+(12 \times \dfrac{1250}{a} )\\C=6a+(\dfrac{15000}{a} )$

For minimum cost (C), the value of C' = 0. Then,

$C'=0\\6-\dfrac{15000}{a^{2}} =0\\a=\sqrt{\dfrac{15000}{6} } \\a=50 \;\rm feet$

And,

$b=\dfrac{1250}{50}\\b= 25 \;\rm feet$

Thus, the dimensions of the rectangle are 50 feet and 25 feet.

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