# Let's say I have $480 to fence in a rectangular garden. The fencing for the north and south sides of the garden costs$10 per foot and the fencing for the east and west sides costs $15 per foot. How can I find the dimensions of the largest possible garden.? Nica2t 2022-08-12 Answered Let's say I have$480 to fence in a rectangular garden. The fencing for the north and south sides of the garden costs $10 per foot and the fencing for the east and west sides costs$15 per foot. How can I find the dimensions of the largest possible garden.?
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Alejandra Blackwell
Let's call the length of the N and S sides x (feet) and the other two we will call y (also in feet)
Then the cost of the fence will be:
$2\cdot x\cdot 10$ for N+S and $2\cdot y\cdot 15$ for E+W
Then the equation for the total cost of the fence will be:
$20x+30y=480$
We separate out the y:
$30y=480-20x\to y=16-\frac{2}{3}x$
Area:
$A=x\cdot y$ replacing the y in the equation we get:
$A=x\cdot \left(16-\frac{2}{3}x\right)=16x-\frac{2}{3}{x}^{2}$
To find the maximum, we have to differentiate this function, and then set the derivative to 0
$A\prime =16-2\cdot \frac{2}{3}x=16-\frac{4}{3}x=0$
Which solves for x=12
Substituting in the earlier equation $y=16-\frac{2}{3}x=8$
Answer: N and S sides are 12 feet, E and W sides are 8 feet, Area is 96 square feet
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