SOLUTION: Can you help: Sue wants to put a rectangular garden on her property using 80 meters of fencing. There is a river that runs through her property so she decides to increase the s

Question 3405: Can you help:
Sue wants to put a rectangular garden on her property using 80 meters of fencing. There is a river that runs through her property so she decides to increase the size of the garden by using the river as one side of the rectangle. Fencing is then needed only on the other three sides.) Let x represent the length of the side of the rectangle along the river. Express the gardens area as a function of x. Find the dimensions of the field so that its area is a maximum.
Found 2 solutions by khwang, pannalal:
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Let the three sides be y, x and y
Sum of the three sides x + 2y = 80 , so y = (80-x)/2
Area of the rectangle A = xy = x(80-x)/2 = 40x -x^2/2

By complete square A = -1/2(x^2 - 80x + (80/2)^2) + 1600/2
= -1/2(x - 40)^2 + 800
Since (x - 40)^2 >=0, -1/2(x - 40)^2 <= 0,
A = -1/2(x - 40)^2 + 800 <= 800,
when x = 40, y = (80-40)/2 = 20.
Hence, when the dimensions of the field are 20,40 and 20 meters
the max area is 800 sq. meters.

Kenny
PS. Also, try to get rid of some redunant words which are not
related to math in this question. Like :
There is a river that runs through her property so she decides to increase the size of the garden by using the river as one side of the rectangle.
Answer by pannalal(9) (Show Source): You can put this solution on YOUR website!
As per question x represent the length of the side of the rectangle
along the river. Let the width of garden be y.
The length of fence = x + 2y = 80
and the area of the garden = xy
We need to maximize the area of the garden.
Suppose x = 40+k where k is some constant either positive or negative
The equation for fencs is

On putting x = 40+k in the above equation:

The area of garden is xy.
On putting the value of x and y in terms of k in area:

To maximize the area, we need to take the value of k = 0.
Then, Area = 800 square meter
and x = 40 and y = 20.

RELATED QUESTIONS
Good morning distinguished Tutors,i kindly need help with the following question-------"A (answered by stanbon)
A farmer has 120 meters of fencing and wants to enclose a rectangular plot of land that... (answered by TimothyLamb)
A farmer has 120 meters of fencing and wants to enclose a rectangular plot of land that... (answered by TimothyLamb)
Farmer Ed has 4,000 meters of fencing, and wants to enclose a rectangular plot that... (answered by macston)
Pls help me with this. It's really hard!! A farmer wants to make a rectangular garden of (answered by bucky,checkley75,Fombitz)
Ely wants to enclose a rectangular garden using 150 meters fencing material. What are the (answered by ikleyn)
Mrs. Thomas also has a garden in her yard and wants to put a 50-meter rectangular fence... (answered by mananth)
Please help me solve this problem: Mrs. Umali wants to make a flower garden such that the (answered by Reese523)
Mr. Johnson used 80 meters of fencing to enclose a rectangular garden. The length of the (answered by josgarithmetic)