.
Since one side is the barn, the rectangle's fence perimeter will be
L + 2W = 60 meters.
Hence, L = 60 - 2W meters.
Area = Length * Width.
Substitute (60-2W) for L:
A = W(60 - 2W) (1)
A = -2W^2 + 60W.
It is a quadratic function. It has the maximum at x = -b/(2a), where "a" is the coefficient at the quadratic term
and "b" is the coefficient at the linear term, according to the general theory.
(See the lessons
- HOW TO complete the square to find the minimum/maximum of a quadratic function
- Briefly on finding the minimum/maximum of a quadratic function
in this site).
In your case, the maximum is at
W = = = 15.
So, W = 15 meters is the width for max area.
Then the length is L = 60 - 2W = 600 - 2*15 = 30 meters.
Find the max area. It is
A = L*W = 30*15 = 450 square meters. It is the maximum area.
Solved.
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My other lessons in this site on finding the maximum/minimum of a quadratic function are
- HOW TO complete the square to find the minimum/maximum of a quadratic function
- Briefly on finding the minimum/maximum of a quadratic function
- HOW TO complete the square to find the vertex of a parabola
- Briefly on finding the vertex of a parabola
- A rectangle with a given perimeter which has the maximal area is a square
- A farmer planning to fence a rectangular garden to enclose the maximal area
- A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
- Finding the maximum area of the window of a special form
- Using quadratic functions to solve problems on maximizing revenue/profit
- OVERVIEW of lessons on finding the maximum/minimum of a quadratic function