Question

17. A rectangular garden 200 square feet in area is to be fenced o against rabbits. Find thedimensions that will require the least amount of fencing given that one side of the garden isalready protected by a barn.18.

Answer 1

Answer:

$x=10$ feet and $y=20$  feet

The least amount of fencing required = 40 feet.

Step-by-step explanation:

Let us take $x$ be the length of the garden and $y$ be width of the garden.

The area of the rectangular garden will be, $A=x y=200$  ...{1}

Perimeter of the garden, $P=2 x+y$.      

[ because one side of the garden is protected by a barn ]

Thus,

$P=2 x+y$

or

$P=2 x+\frac{200}{x}$          [using 1]

$\frac{d P}{d x}=2-\frac{200}{x^{2}}$

For points of maxima or minima

$\frac{d P}{d x}=0 \Rightarrow 2-\frac{200}{x^{2}}=0 \Rightarrow x^{2}=100 \Rightarrow x=\pm 10$

Since length cannot be negative, $x=-10$ is rejected.

Thus,

$\frac{d^{2} P}{d x^{2}}=-(-2) \frac{200}{x^{3}}=\frac{400}{x^{3}}$

$\left(\frac{d^{2} P}{d x^{2}}\right)_{x=10}=\frac{400}{10^{3}}>0$

Hence at $x=10,$, The perimeter $(P)$ will be min.

On substituting $x=10$ in $x y=200,$ then $y=20$

Hence, the dimensions of the garden are:

$x=10$ feet and $y=20$  feet

The perimeter, $P=20+20=40$ feet i.e the least amount of fencing required.