SOLUTION: Nicky has 120 feet of fence to put around a rectangular garden. if a 10 foot opening is left on one side for a gate. what would be the length and width for maximum area? L=20 ft w=

Question 1001420: Nicky has 120 feet of fence to put around a rectangular garden. if a 10 foot opening is left on one side for a gate. what would be the length and width for maximum area? L=20 ft w=20 ft
Answer by josgarithmetic(39618) (Show Source): You can put this solution on YOUR website!
A square shape will give maximum area. I do not give that analysis here. The perimeter of this would be , to include both the fence and the gate(unfenced). Cut the 130 feet into four equal parts, and this is the side of the square garden.

feet square.

--
MAXIMIZE AREA

Dimensions are x and y.
The entire perimeter of the garden is length_of_fencing PLUS 10 feet for gate; so this means the perimeter of the garden is feet.

Two basic equations are needed.
.

Use these to make a function A dependent on just ONE of the variables, either x or y.

------Area A is a function of the dimension, x, a quadratic function, and you can look for the MAXIMUM. You know there is a maximum because the coefficient on will be negative...
The max area will occur in the exact middle of the roots or zeros or x-intercepts of .
-
Solve for the x-intercepts.

The roots are 0 and 65.
The value for x exactly in the middle is .

WHAT IS THE VALUE FOR y FOR THIS VALUE OF x ?
WHAT IS THE AREA AT THIS VALUE OF x ?
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