.
Since one side is the river, the rectangle's fence perimeter will be
L + 2W = 1400.
Hence, L = 1400 - 2W.
Area = Length * Width.
Substitute (1400-2W) for L:
A = W(1400 - 2W)
A = -2W^2 + 1400W.
This is a quadratic function. It has the maximum at x = -b/(2a), 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).
For our quadratic function the maximum is at
W = = = 350.
So, W = 350 ft is the width for max area.
Then the length is L = 1400 - 2W = 1400 - 2*350 = 700 ft.
Find the max area. It is
A = L*W = 700*350 = 245000 square feet.
The plot of the quadratic function for the area is shown below: y = area and x = width.
See my lessons in this site on finding the maximum/minimum of a quadratic function
- 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