Question

I would like to create a rectangular orchid garden that abuts my house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $20per foot, and the fencing for the east and west sides costs $10 per foot. If I have a budget of $200 for the project, what are the dimensions of the garden with the largest area I can enclose? HINT [See Example 2.]

ft (smaller value) ? ft (larger value)
ft2

Answer 1

Length =
`5\ \text{ft}`

Breadth =
`5\ \text{ft}`

Explanation:

Let
`x` be the length of the garden

and
`y` be the width of the garden

From the details of the cost in the question we get

`20x+2* (10y)=200\\\Rightarrow 20x+20y=200\\\Rightarrow x+y=(200)/(20)\\\Rightarrow x+y=10\\\Rightarrow y=10-x`

Now area of the garden is

`A=xy\\\Rightarrow A=x(10-x)\\\Rightarrow A=10x-x^2`

Differentiating with respect to x we get

`(dA)/(dx)=10-2x`

Equating with 0

`0=10-2x\\\Rightarrow x=(-10)/(-2)\\\Rightarrow x=5`

Double derivative of the area is

`(d^2A)/(dx^2)=-2<0`

So, area is maximum at
`x=5`

`y=10-x=10-5\\\Rightarrow y=5`

So, the length and breadth of the rectangle is
`5\ \text{ft}`.