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g The fencing of the left border costs $10 per foot, while the fencing of the lower border costs $2 per foot. (No fencing is required along the river.) You want to spend $520 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose?

1 Answer

1 vote

Answer:


Area = 1690ft^2\\x = 130ft, y = 26ft

Explanation:

Given the costs we can form an equation:


10y + 2x =520 - Eq(A)

and fencing is triangular such that the area enclosed can be written as:


A = (xy)/(2) -Eq(B)

  • First need to convert the above equation so that it is only in terms of one variable. [either x or y]

To make the equation only in terms of
x we can substitute
y from Eq(A) i.e,
y =(520-2x)/(10), to Eq(B)


A = (x)/(2) (520-2x)/(10)

simplify


A = (520x - 2x^2)/(20)


A = -(1)/(10)x^2 + 26x

  • Now, in order to find the maximum area enclosed we can find
    (dA)/(dx) and equate it zero.


(dA)/(dx) = -(1)/(5)x + 26


0 = -(1)/(5)x + 26


-26 = -(1)/(5)x


x = 130

we have the length of one dimension: specifically, the lower fence will be
x =130ft

we can use this value of
x to find the corresponding value of
y. From Eq(A)


10y + 2x =520


10y +2(130) = 520


y = (520 - 2(130))/(10)


y = 26

the length of the left fence will be
y =26ft

  • The enclosed area by the fence will be


A = (xy)/(2)


A = ((130)(26))/(2)


A = 1690

Hence the maximum area that can enclosed by the fences provided the costs will be
1690ft^2

  • You can even check the cost of the dimensions whether they all add up to $520 or not.

Use Eq(A)


10y + 2x =520


10(26) + 2(130) =520

and indeed it does!

User Krirk
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