Question

A rectangular garden must have an area of 64 square feet. find the minimum perimeter of the garden.

Answer 1

A rectangular garden must have an area of 64 square feet then the minimum perimeter of the garden is 32 feet and this can be determined by using the formula of the perimeter of a rectangle.

Given :

A rectangular garden must have an area of 64 square feet.

The area of a rectangle is given by:

$\rm A =l\times w$

where 'l' is the length of the rectangle and 'w' is the width of the rectangle.

Given that area of the rectangular garden is 64 square feet that is:

64 = lw

$\rm w = \dfrac{64}{l}$   ---- (1)

Now the perimeter of a rectangle is given by:

P = 2(l + w)

Put the value of w in the above equation.

$\rm P = 2 (l + \dfrac{64}{l})$   ---- (2)

For minimum perimeter differentiate the above equation with respect to the length of the garden.

$\rm P' = 2 - \dfrac{128}{l^2}$  

Now, equate the above equation to zero.

$\rm 0 = 2-\dfrac{128}{l^2}$

$l^2 = 64$

$l = 8$

Now put the value of l in equation (2).

$\rm P = 2(8 + \dfrac{64}{8})$

P = 32 feet.

A rectangular garden must have an area of 64 square feet then the minimum perimeter of the garden is 32 feet.

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Answer 2

The minimum value of perimeter for rectangular garden   = 32 feet.

If $l$ is the length of rectangle and $w$ is the width of rectangle than ,

Area of rectangle is given by  equation = $l\times w$ .........(1)

also the perimeter  P  of the rectangle is given by =  $2\times (l+ w)$ .........(2)

Given , Area of a rectangular garden = 64 square feet

on putting the value in  equation (1)

we get ,

64 = $l\times w$

so we can get ,

$\dfrac{64}{w} = l$...........(3)

From equation (2) and (3) we get

P = $2 \times( \dfrac{64}{w}+w )$....... (4)

The function $2 \times( \dfrac{64}{w}+w )$. attains its minima at  $w$ = 8 the value of which can be found out  putting by putting  

so P = $2 \times (\dfrac{64}{8} +8)= 32$

So The minimum value of perimeter  P  = 32 feet.

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