Given conditions:
Total spending limits = 400$
Left border cost = 8$ per foot
Lower border cost = 2$ per foot
Area of the garden = A = xy/2
Required answers:
Dimension of the garden
&
Area enclosed by the garden
Solution:
Let suppose
The left border of the garden = y
The lower border of the garden = x
According to given conditions;
400 = 2x + 8y -------------------------------------------------- (1)
Where left side (400) represents the total spending limits on the garden and right side represents the total cost on the left and lower border.
Solving the equation;
2x = 400 – 8y
--------------------------------------------------- (2)
x = (400 – 8y) / 2
x = 200 – 4y ----------------------------------------------------- (3)
As area is given as
A = xy/2 ---------------------------------------------------------- (4)
Putting equation (3) in equation (4), we get
A = ((200 – 4y) * y) /2
A = (200y – 4 y^2 ) / 2
Or
A = 100y – 2y^2 ----------------------------------------------- (5)
To get the maximum garden area, take the derivative of equation (5) and make it equal to “0”;
dA/dy = d/dy (100y – 2y^2)
dA/dy = 100 – 4y
Now
100 – 4y = 0
4y = 100
y = 25
Putting value of y = 25 in equation in equation (1);
400 = 2x + 8(25)
400 = 2x + 200
2x = 400 – 200
2x = 200
x = 100
Thus the dimensions of the garden are as follow ;
Left border length = 25 foot
Lower border length = 100 foot
While the right border length can be calculated by Pythagoras theorem, as the garden is in shape of a right angled triangle;
Let the right side = z
Then z = square root of (x^2 + y^2)
z = square root of ((100)^2 +(25)^2)
z = square root of (10000 + 625 )
z = square root of ( 10625 )
z = 103 foot (approximately)
now the area of the garden is given as ;
A = xy/2
Putting values;
A = (100 * 25) / 2
So area of the garden is calculated as;
A = 1250 foot^2
