Question

A room 5 m long and 4 m wide is surrounded by a verandah. If the verandah occupies an area
of 22 m. find the width of the varandah.​

Answer 1

Given :-

Length of the room = 5 m

Width of the room = 4 m

Area of the verandah = 22 m

To Find :-

The width of the verandah.

Analysis :-

Consider the width of the verandah as a variable.

Using the variables, make the length and the breadth of the verandah.

Then substitute the values in Area of verandah = Area of outer rectangle - Area of room

Next find the value of the variable accordingly.

Solution :-

We know that,

• l = Length
• b = breadth
• a = Area

By the formula,

$\underline{\boxed{\sf Area \ of \ rectangle=Length \times breadth}}$

Given that,

Length (l) = 5 m

Breadth (b) = 4 m

Substituting their values,

Area = 5 × 4

Area = 20 m²

Therefore, the area of the room is 20 m².

Let the width of the verandah be 'x'.

Dimensions of verandah,

Length of the verandah = 5 + 2x

Breadth of the verandah = 4 + 2x

∴ Area of verandah = Area of verandah − Area of room

Making an equation,

$\sf 22 = 4x^2 + 18x + 20 - 20$

$\sf 22 = 4x^2 + 18x$

$\sf 11 = 2x^2 + 9x$

$\sf 2x^2 + 9x - 11 = 0$

$\sf 2x^2 + 11x -2x - 11 = 0$

$\sf x(2x + 11) - 1(2x + 11) = 0$

$\sf (x - 1) (2x +11) = 0$

$\sf x = 1$

Now,

$\sf 2x + 11 = 0$

Transposing 11,

$\sf 2x=0-11$

$\sf 2x=-11$

Finding the value of x,

$\sf x=-\dfrac{11}{2}$

Since the value cannot be negative,

x = 1 meter

Therefore, the width of the verandah is 1 m.

Answer 2

Given data : A room 5 m long and 4 m wide is sorrounded by a verandah. The verandah occupies an area of 22 m².

Solution : Assume that the width of the verandah is the same in all directions. Here we take the width of the verandah to be x.

The verandah occupies an area of 22 m². ----{1}

➜ Length of the verandah (with room) = (5 + 2x) m

➜ Breadth of the verandah (with room) = (4 + 2x) m

Now,

➜ Area of the verandah (with room)

= length * breath

➜ Area of the verandah (with room)

= (5 + 2x) * (4 + 2x)

➜ Area of the verandah (with room)

= 20 + 10x + 8x + 4x²

➜ Area of the verandah (with room)

= 4x² + 18x + 20

Now, a/c to given data;

➜ Length of the room = 5 m

➜ Breadth of the room = 4 m

Let, shape of the room be rectangular,

➜ Area of the room = length * breadth

➜ Area of the room = 5 * 4

➜ Area of the room = 20 m²

Here, we know that, (a/c to figure)

➜ Area of the verandah (with room) = Area of the room + Area of the verandah

➜ 4x² + 18x + 20 = 20 + 22 [from {1}]

➜ 4x² + 18x + 20 = 42

➜ 4x² + 18x + 20 - 42 = 0

➜ 4x² + 18x - 22 = 0

Divide eq. by 2

➜ 2x² + 9x - 11 = 0

➜ 2x² + 11x - 2x - 11 = 0

➜ x (2x + 11) - 1 (2x + 11) = 0

➜ (x - 1) (2x + 11) = 0

➜ x - 1 = 0 or 2x + 11 = 0

➜ x = 1 or 2x = - 11

➜ x = 1 or x = - 11/2

Here, we know, width of the verandah is never negative. Hence, x ≠ - 11/2 and x = 1

Answer : Hence, the width of the verandah is 1 m.