Question

the perimeter of a rectangular field is 100m.if the area of the field is 600 sq.m find the dimensions of the field.​

Answer 1

Given

• Perimeter of a rectangular field = 100m
• Area of rectangular field= 600m²

Find out

• Dimensions of the field

Solution

★Let the length of rectangular field be x and its breadth be y

➟ Perimeter of field = 100 m

➟ 2(x + y) = 100

➟ x + y = 50

➟ x = (50 - y) ----(i)

Now,

➟ Area of field = 600 m²

➟ xy = 600

Put the value of x

➟ (50 - y)y = 600

➟ 50y - y² = 600

➟ - y² + 50y - 600 = 0

★ Split middle term ★

➟ - y² + 30y + 20y - 600 = 0

➟ - y(y - 30) + 20(y - 30) = 0

➟ (y - 30)(- y + 20) = 0

Either

➟ (y - 30) = 0

➟ y = 30 m

Or

➟ (- y + 20) = 0

➟ - y = - 20

➟ y = 20 m

Put the values of y in eqⁿ (i)

★ Case : 1

• y = 30 m

➟ x = (50 - y)

➟ x = 50 - 30

➟ x = 20 m

★ Case : 2

• y = 20 m

➟ x = (50 - y)

➟ x = 50 - 20

➟ x = 30 m

Hence,

• Length of field = 20 m or 30 m
• Breadth of field = 30 m or 20 m

Answer 2

Given that ,

• Perimeter of rectangle = 100 m
• Area of rectangle = 600 m²

Let , length and breadth of a rectangle be " x " And " y "

Thus ,

2(x + y) = 100 --- (i)

xy = 600 --- (ii)

From eq (i) and eq (ii) , we get

$\sf \mapsto 2( \frac{600}{y} + y ) = 100 \\ \\ \sf \mapsto 600 + {(y)}^{2} = 50y \\ \\ \sf \mapsto {(y)}^{2} - 50y + 600 = 0 \\ \\ \sf \mapsto {(y)}^{2} - 30y - 20y + 600 = 0 \\ \\ \sf \mapsto y(y - 30) - 20(y - 30) = 0 \\ \\ \sf \mapsto (y - 20)(y - 30) = 0 \\ \\ \sf \mapsto y = 20 \: \: or \: \: y = 30$

Put the values of y in eq (i) , we get

$\sf \mapsto2(x + 20) = 100 \: or \: 2(x + 30) = 100 \\ \\ \sf \mapsto </p><p>x + 20 = 50 \: or \: x + 30 = 50 \\ \\ \sf \mapsto</p><p>x = 30 \: or \: x = 20$

$\therefore \sf \underline{The \: length \: and \: brea dth \: of \: a \: rectangle \: are \: </p><p>30 \: cm \: or \: 20 \: cm \: and \: 20 \: or \: 30 \: cm}$