Question

Perimeter (km)	Length of rectangle (km)	Width of rectangle (km)	Area of protected region ( $k m^{2}$ )
$20$	$1$
$20$	$2$
$20$
$20$
$20$	$5$
$20$
$20$
$20$	$9.8$

Calculate the area with other values of the perimeter. Which shape seems to maximize the area of the protected region?

Accepted Answer

Take any value to find the area of protected region.

Area of rectangle $= l \times b$

Perimeter of rectangle $= 2 (l + b)$

Perimeter is constant i.e $20 k m$

Now,

$(1)$ If $L = 1 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (1 + b) \Rightarrow 20 = 2 + 2 b \Rightarrow \frac{20 - 2}{2} = b \Rightarrow 9 = b$

Area of protected region $= l \times b = 1 \times 9 = 9 k m^{2}$

$(2)$ If $L = 2 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (2 + b) \Rightarrow 20 = 4 + 2 b \Rightarrow \frac{20 - 4}{2} = b \Rightarrow 8 = b$

Area of protected region $= l \times b = 2 \times 8 = 16 k m^{2}$

$(3)$ If $L = 3 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (3 + b) \Rightarrow 20 = 6 + 2 b \Rightarrow \frac{20 - 6}{2} = b \Rightarrow 7 = b$

Area of protected region $= l \times b = 3 \times 7 = 21 k m^{2}$

$(4)$ If $L = 4 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (4 + b) \Rightarrow 20 = 8 + 2 b \Rightarrow \frac{20 - 8}{2} = b \Rightarrow 6 = b$

Area of protected region $= l \times b = 4 \times 6 = 24 k m^{2}$

$(5)$ If $L = 5 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (5 + b) \Rightarrow 20 = 10 + 2 b \Rightarrow \frac{20 - 10}{2} = b \Rightarrow 5 = b$

Area of protected region $= l \times b = 5 \times 5 = 25 k m^{2}$

$(6)$ If $L = 6 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (6 + b) \Rightarrow 20 = 12 + 2 b \Rightarrow \frac{20 - 12}{2} = b \Rightarrow 4 = b$

Area of protected region $= l \times b = 6 \times 4 = 24 k m^{2}$

$(7)$ If $L = 7 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (7 + b) \Rightarrow 20 = 14 + 2 b \Rightarrow \frac{20 - 14}{2} = b \Rightarrow 3 = b$

Area of protected region $= l \times b = 7 \times 3 = 21 k m^{2}$

$(6)$ If $L = 9.8 k m$

Perimeter $= 2 (l + b)$

$\Rightarrow 20 = 2 (9.8 + b) \Rightarrow 20 = 19.6 + 2 b \Rightarrow \frac{20 - 19.6}{2} = b \Rightarrow 0.2 = b$

Area of protected region $= l \times b = 9.8 \times 0.2 = 1.96 k m^{2}$

Perimeter (km) Length of rectangle (km) Width of rectangle (km) Area of protected region ( $k m^{2}$ )

$20$ $1$ $9$ $9$

$20$ $2$ $8$ $16$

$20$ $3$ $7$ $21$

$20$ $4$ $6$ $24$

$20$ $5$ $5$ $25$

$20$ $6$ $8$ $16$

$20$ $7$ $3$ $21$

$20$ $9.8$ $0.2$
$1.96$

Area of protected region will maximize when length and breadth both will be $5 k m$

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