Question

A room is half as long again as it is broad. Cost of Carpeting the room at Rs 3.25 per m^2 is Rs 175.5 and cost of papering the walls at Rs 1.40 per m^2 is Rs 240.89. If 1 Door and 2 windows occupy 8m^2,Find dimensions of room. ​

Answer 1

Solution :

Let the breadth of the room be b metre. Then,

$\sf \: Length = (b + \dfrac{b}{2})m = ( \dfrac{3b}{2} )m$

$\bigstar \boxed{ \purple{ \sf \: Area \: of \: Rectangle = Length \times Breadth}}$

$\longrightarrow \sf \: ( \dfrac{3b}{2} \times b) {m}^{2}$

$\longrightarrow \sf \: (\dfrac{ {3b}^{2} }{2} ) {m}^{2}$

Cost of Carpeting the room at the rate of Rs 3.25 per m^2 :

$\longrightarrow \sf Rs( \dfrac{ {3b}^{2} }{2} \times 3.25)$

Given Cost of Carpeting is Rs 175.5

$\therefore \sf \: \dfrac{ {3b}^{2} }{2} \times 3.25 = 175.5$

$\longrightarrow \sf \: {b}^{2} = \dfrac{175.5 \times 2}{3 \times 3.25}$

$\longrightarrow \sf \: {b}^{2} = \dfrac{351}{6.5}$

$\longrightarrow \sf \: {b}^{2} = \cancel\dfrac{3510}{65}$

$\longrightarrow \sf \: {b}^{2} = 36$

$\longrightarrow \sf \: {b} = \sqrt{36}$

$\longrightarrow \sf \: \pink{ {b} = 6 \: m}$

Breadth = 6 m

$\sf \: Length = ( \dfrac{3 \times 6}{2} )m$

$\sf \longrightarrow \: l = \cancel\dfrac{18}{2}m$

$\sf\longrightarrow \pink{\: l = 9m}$

Let the Height of the room be h metres. Then,

$\bigstar \boxed{ \purple{ \sf \: Area \: of \: 4 \: walls =2 ( Length + Breadth) \times Height }}$

$\longrightarrow \sf \: 2(6 + 9) \times h \: {m}^{2}$

$\longrightarrow \sf \: 12 + 18 \: h \: {m}^{2}$

$\longrightarrow \sf \: 30 \: h \: {m}^{2}$

Given 1 Door and 2 Windows occupy 8 m^2

$\bigstar \boxed{ \purple{ \sf \: Required \: Area = Area \: of \: 4 \: walls \: - \: Area \: of \: 1 \: door \: and \: 2 \: Windows}}$

$\longrightarrow \sf \: 30h - 8 {m}^{2}$

Cost of papering walls at Rs 1.4 per m^2 :

$\longrightarrow \sf \: 30h - 8 \times 1.4$

Given Cost of Papering is Rs 240.8

$\therefore \sf \: 30h - 8 \times 1.4 = 240.8$

$\longrightarrow \sf \: 30h - 8 = \dfrac{240.8}{1.4}$

$\longrightarrow \sf \: 30h - 8 = \cancel\dfrac{24080}{140}$

$\longrightarrow \sf \: 30h - 8 = 172$

$\longrightarrow \sf \: 30h = 172 + 8$

$\longrightarrow \sf \: 30h =180$

$\longrightarrow \sf \: h = \cancel\dfrac{180}{30}$

$\sf\longrightarrow \pink{\: h = 6m}$

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Hence, Dimensions of the room are :

• Length = 9 m

• Breadth = 6 m

• Height = 6 m

Answer 2

Given :

• A room is half as long as it is broad.
• Cost of Carpeting the room at Rs 3.25 per m^2 is Rs 175.5.
• Cost of papering the wall at Rs. 1.40 per m² is ₹ 240.89.
• 1 door and 2 windows occupies 8 m²

To Find :

• Dimensions of the room.

Solution :

Let the broadness of the room be x m.

Case 1 :

$\mathtt{Length\:of\:the\:room\:=\:\dfrac{x}{2}+x}$ ___(1)

Area of the room :

Formula :

$\large{\boxed{\mathtt{\red{Area_{room}\:=\:length\:\times\:breadth}}}}$

Block in the data,

$\longrightarrow$ $\mathtt{Area_{room}\:=\:\Big(\dfrac{x}{2}+x\Big)\:\times\:x}$

$\longrightarrow$ $\mathtt{\dfrac{x+2x}{2}\:\times\:x}$

$\longrightarrow$ $\mathtt{\dfrac{3x}{2}\:\times\:x}$

$\longrightarrow$ $\mathtt{\dfrac{3x^2}{2}}$

$\large{\boxed{\sf{\red{Area_{room}\:=\:\dfrac{3x^2}{2}\:m^2}}}}$

Cost Of Carpeting :

We have been given the cost of carpeting the room as ₹ 3.25 per m²

•°• Cost of carpeting the room of area of 3x²/2 m² will be the product of area of the room and the cost of carpeting the room at 3.25 per m² which is ₹ 175.5

$\longrightarrow$ $\mathtt{Cost\:of\:carpeting\:=\:3.25\:\times\:\dfrac{3x^2}{2}}$

$\longrightarrow$ $\mathtt{175.5\:=\:\dfrac{9.75\:x^2}{2}}$

$\longrightarrow$ $\mathtt{175.5\:=\:\cancel{\dfrac{9.75\:^{4.875}}{2}x^2}}$

$\longrightarrow$ $\mathtt{175.5\:=\:4.875x^2}$

$\mathtt{\cancel\dfrac{175.5}{4.875}\:=\:x^2}$

$\longrightarrow$ $\mathtt{36\:=\:x^2}$

$\longrightarrow$ $\mathtt{\sqrt{36}\:=x}$

$\longrightarrow$ $\mathtt{x=6}$

$\large{\boxed{\sf{\purple{Breadth\:of\:the\:room\:=\:6\:m}}}}$

Substitute, x = 6 in equation (1)

$\longrightarrow$ $\mathtt{\dfrac{x}{2}\:+x}$

$\longrightarrow$ $\mathtt{\cancel{\dfrac{6\:^3}{2^1}}+6}$

$\longrightarrow$ $\mathtt{3\:+\:6}$

$\longrightarrow$ $\mathtt{9}$

$\large{\boxed{\sf{\red{Length\:of\:the\:room\:=\:9\:m}}}}$

Area of papering :

Papering the walls at ₹ 1.40 per m² is ₹ 240.89

Let's find the area of the four walls.

Excluding the ceiling and the floor of the room, we will find the surface area of the room using the formula of surface area of the cuboid, since the room is in cuboidal shape.

Formula :

$\large{\boxed{\sf{\blue{Area\:=\:2(l+b)h}}}}$

Where,

• l = length = 9 m
• b = breadth = 6
• h = height = ?

Block in the data,

$\longrightarrow$ $\mathtt{Area_{4\:walls}\:=\:2\:(9\:+6)\:h}$

$\longrightarrow$ $\mathtt{Area_{4\:walls}\:=\:2\:(15)\:h}$

$\longrightarrow$ $\mathtt{Area_{4\:walls}\:=\:30\:h}$

$\large{\boxed{\sf{\green{Area\:_{4\:walls\:=\:30\:h\:n^2}}}}}$

Given that, the area occupied by 1 door and 2 windows is 8 m²

•°• The area available for papering of the four walls will be the difference between 8 m² and area of 4 walls.

Cost of papering at the walls at the rate of ₹ 1.40 is ₹ 240.89

$\longrightarrow$ $\mathtt{Cost_{papering}\:=\:(30h-8)\:\times\:1.40}$

$\longrightarrow$ $\mathtt{\cancel{\dfrac{240.80}{1.40}}\:=\:(30h-8)}$

$\longrightarrow$ $\mathtt{172\:=\:30h-8}$

$\longrightarrow$ $\mathtt{172+8=30h}$

$\longrightarrow$ $\mathtt{180=30h}$

$\longrightarrow$ $\mathtt{\cancel{\dfrac{180}{30}}+h}$

$\longrightarrow$ $\mathtt{6=h}$

$\large{\boxed{\sf{\red{Height\:of\:the\:room\:=\:6\:\:m}}}}$