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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Answers (1)

Alejandra Blackwell
Answered 2022-08-13 Author has 14 answers
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 x $ 10 for N+S and 2 y $ 15 for E+W
Then the equation for the total cost of the fence will be:
20 x + 30 y = 480
We separate out the y:
30 y = 480 - 20 x y = 16 - 2 3 x
Area:
A = x y replacing the y in the equation we get:
A = x ( 16 - 2 3 x ) = 16 x - 2 3 x 2
To find the maximum, we have to differentiate this function, and then set the derivative to 0
A = 16 - 2 2 3 x = 16 - 4 3 x = 0
Which solves for x=12
Substituting in the earlier equation y = 16 - 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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