Question

My friend has $672 to spend on a fence for her rectangular garden. She wants to use cedar fencing which costs $17/yard on one side, and cheaper metal fencing which costs $7/yard for the other three sides.

What are the dimensions of the garden with the largest area she can enclose?

Answer 1

The dimensions of the garden with the largest area she can enclose are given by 24*14 square yards.

The area (A) of the rectangle with length L and Width W is given by,

A=L*W

The maximum of $y=ax^2+bx$ occurs at x = -b/2a

Let the length and width of the garden be L and W respectively.

Now let my friend use cedar fencing for one width and cheaper metal fencing for rest sides.

Rate of cedar fencing is $17/yard

Then she has to pay for cedar fencing = 17W

rate of cheaper metal fencing is $7/yard

Then she has to pay for metal fencing = 7W+7*2L = 7W+14L

Then according to condition,

17W+7W+14L = 672

24W+14L = 672

12W+7L = 336

7L = 336-12W

L = (336-12W)/7

Then the area of the rectangular garden is given by,

A = L*W

$A=\frac{336-12W}{7}\times W\\A=-\frac{12}{7}W^2+48W$

So here a = -12/7 and b = 48

Then maximum of W occurs at,

W = -48/(2*(-12/7)) = (48*7)/(2*12) = 336/24 = 14

Then maximum width = 14 yards

then length (L) = (336-12*14)/7 = 168/7 = 24 yards

Hence the dimensions with the leargest area she can enclose is given by = 24 * 14 square yards.

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