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

Answer:

Length = $5\ \text{ft}$

Breadth = $5\ \text{ft}$

Step-by-step 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

$20x+2\times (10y)=200\\\Rightarrow 20x+20y=200\\\Rightarrow x+y=\dfrac{200}{20}\\\Rightarrow x+y=10\\\Rightarrow y=10-x$

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

$\dfrac{dA}{dx}=10-2x$

Equating with 0

$0=10-2x\\\Rightarrow x=\dfrac{-10}{-2}\\\Rightarrow x=5$

Double derivative of the area is

$\dfrac{d^2A}{dx^2}=-2$

So, area is maximum at $x=5$

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

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