Alandscape architect wished to enclose a rectangular garden on one side by a brick wall costing $20/ft and on the other three sides by a metal fence costing $10/ft. if the area of the garden is 122 square feet, find the dimensions of the garden that minimize the cost.

The dimensions of the garden that minimize the cost is 9.018 feet(length) and 13.528 feet(width)

Step-by-step explanation:

Let the length of garden be x

Let the breadth of garden be y

Area of Rectangular garden =

We are given that the area of the garden is 122 square feet

So, ---A

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $20/ft

So, cost of brick along length x = 20 x

On the other three sides by a metal fence costing $10/ft.

So, Other three side s = x+2y

So, cost of brick along the other three sides= 10(x+2y)

So, Total cost = 20x+10(x+2y)=20x+10x+20y=30x+20y

Total cost = 30x+20y

Substitute the value of y from A

Total cost =

Total cost =

Now take the derivative to minimize the cost

Equate it equal to 0

Now check whether it is minimum or not

take second derivative

Substitute the value of x

Since it is positive ,So the x is minimum

Now find y

Substitute the value of x in A

Hence the dimensions of the garden that minimize the cost is 9.018 feet(length) and 13.528 feet(width)