Mathematics, 01.09.2019 02:30 caghwhw

Arectangular garden of area 832 square feet is to be surrounded on three sides by a brick wall costing $8 per foot and on one side by a fence costing $5 per foot. find the dimensions of the garden such that the cost of the materials is minimized.

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Answer from: dontcareanyonemo

The dimensions of the garden such that the cost of the materials is minimized are 26 feet, 32 feet.

Let the length and the breadth of the rectangular garden be x and y respectively.

Area of a rectangle =

832 square feet=

⇒ $x=\frac{832}{y}$

Assume the side y is a fence, and the rest brick.

Total cost, $C(y) = $5y + $8y + 2.8\frac{832}{y}$

⇒ $C(y)=13y+\frac{13312}{y}$

Differentiate the cost with respect to y and equate the derivative C'(y) to zero for maximum or minimum value of y.

$C'(y)=13-\frac{13312}{y^{2} } \\0=13-\frac{13312}{y^{2} } \\$

$y=\sqrt{\frac{13312}{13} } \\y=32$

Therefore the length of the rectangular garden is

$x=\frac{832}{32} \\=26$

And the second derivative is

$C''(y)=\frac{26624}{y^{3} }\\C''(32)=\frac{26624}{32^{3} }\geq 0$

So the cost is minimum at y=32 feet

Therefore the dimensions of the garden are x=26 feet, y=32feet.

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Answer from: ericka79

Nice problem!

Area of a rectangle = length * width, or length = Area / width
Let the Width be W
then the length is 832/W
Assume the side W is a fence, and the rest brick.
Total cost, C(W) = $5*W + $8W + 2*$8*(832/W)
Simplifying, C(W)=13W+13312/W

To find the minimum cost, we differentiate the cost with respect to W, and equate the derivative C'(W) to zero.
then C'(W)=13-13312/W^2=0
Rewrite C'(W) as (13W^2-13312)/(W^2)=0, we solve for W and get
W=sqrt(13312/13)=32
Therefore length, L=832/32=26
Check L*W=26*32=832 ok [ note L>W, because this costs less $]
Check L=26 and W=32 is the minimum (as opposed to maximum),
we calculate C"(W)=26624/W^3 > 0 which means that W=26 is a minimum for C(W).
So the dimensions of the garden are 26' x 32', with fence on the W=32' dimension.
Just by curiosity, total cost = C(32)=$832, and average cost = $7.17/'
all sound reasonable.

Answer from: Quest

On monday,mrs.wise bought 3 lbs of apples at $4per lb= 4*3=$12on tuesday ,20%off and she buy 5 lbs of apples = $4-20%=3.20*5lb=$16example: other wise when she buy 5lbs on monday she have to spend 4*5=20

Answer from: Quest

The answer to the question is d

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