Question

$\mathfrak{\huge{\underline{Question:}}}$

The length of a room is 50% more than its breadth. The cost of carpeting the room at rate of Rs.38.50m^2 is Rs 924 and the cost of painting the walls at the rate of Rs 5.50 per m^2 is Rs 1320. Find the dimensions of the room.

● Content Quality answers required,All the best :D​

Answer 1

Answer-

Dimensions of the room = 6m (length), 4m (breadth) and 12m (height)

$\rule{100}2$

Explanation-

Let the breadth of the room be x m and length be y m.

The length of a room is 50% more than it's breadth.

$\implies\:\sf{y\:=\:x\:+\:50\%\:of\:x}$

$\implies\:\sf{y\:=\:x\:+\:\frac{50x}{100}}$

$\implies\:\sf{y\:=\:x\:+\:\frac{x}{2}}$

$\implies\:\sf{y\:=\:\frac{2x\:+\:x}{2}}$

$\implies\:\sf{y\:=\:\frac{3x}{2}}$

The cost of carpeting the room at rate of Rs.38.50m² is Rs 924.

Now,

The cost of carpeting the room at1m² costs Rs. 38.50

Rs. 38.50 cost is the carpeting of room at 1 m²

For Rs. 1 = $\dfrac{1}{38.50}$

Total Rupees = 924

For Rs. 924 it will be $\dfrac{924}{38.50}$

$\sf{Area \:of\: floor\:=\:\frac{Cost\:of\:carpeting\:the\:floor\:at\:the\:rate\:of\:Rs.38.50m^2}{Cost\:of\:carpeting\:the\:floor}}$

$\implies\:\sf{\dfrac{924}{38.50}}$

$\implies\:\sf{\dfrac{9240}{385}}$

$\implies\:\sf{\dfrac{168}{7}}$

$\implies\:\sf{24}$

We know that floor is in rectangular shape.

So, Area of rectangle = Area of floor

Area of floor = length × breadth

$\implies\:\sf{24m\:=\:x\:\times\:y}$

$\implies\:\sf{24\:=\:x\:\times\:\frac{3x}{2}}$

$\implies\:\sf{24\:=\:\frac{3x^2}{2}}$

$\implies\:\sf{48\:=\:3x^2}$

$\implies\:\sf{\frac{48}{3}\:=\:x^2}$

$\implies\:\sf{16\:=\:x^2}$

$\implies\:\sf{\sqrt{16}\:=\:x}$

$\implies\:\sf{4\:=\:x}$

$\implies\:\sf{x\:=\:4}$

•°• Breadth of the room is 4 m

Substitute value of x = 4 in y.

$\implies\:\sf{y\:=\:\frac{3(4)}{2}}$

$\implies\:\sf{y\:=\:\frac{12}{2}}$

$\implies\:\sf{y\:=\:6}$

•°• Length of the room is 6 m

We have to find the dimensions of the room. Means, length, breadth and height of the room.

From above calculations we have length and breadth of the room.

Now,

The cost of painting the walls at the rate of Rs 5.50 per m² is Rs 1320.

$\sf{Area\:of\:four\:walls\:\dfrac{1320}{5.50}}$

$\implies\:\sf{\dfrac{13200}{55}}$

$\implies\:\sf{\dfrac{2640}{11}}$

$\implies\:\sf{240}$

A wall has length, breadth and height too.

Area of four walls = 2(length + breadth) × height

Substitute the known values above

$\implies\:\sf{240\:=\:2(6\:+\:4)h}$

$\implies\:\sf{120\:=\:10h}$

$\implies\:\sf{\frac{120}{10}\:=\:h}$

$\implies\:\sf{h\:=\:12}$

•°• Height of the room is 12 m.

Answer 2

$\huge\star{\pink{\underline{\mathfrak{Answer :-}}}}$

Height = 12 m

Breadth = 4 m

Length = 6 m

$\huge\star{\pink{\underline{\mathfrak{Explanation :-}}}}$

Let us take the breadth as b.

According to the question, length is 50% more than breadth.

So, length ( l ) = b + 50% of b

$\leadsto {l = b + (\dfrac{50}{100} \times b)}$

$\leadsto {l = b + \dfrac{b}{2}}$

$\leadsto {l = \dfrac{3b}{2}}$

Now,

Rate of carpenting the room given = Rs. 38.50 per m^2

And the cost = Rs. 924

So, Area of the room = $\dfrac{cost}{rate}$

So,

$\leadsto {Area = \dfrac{924}{38.50}}$

$\leadsto {Length \times breadth = 924 \div \dfrac{385}{100} }$

$\leadsto { \frac{3b}{2} \times b= 924 \times \dfrac{100}{385}}$

$\leadsto {\dfrac{3 {b}^{2} }{2} = 24}$

$\leadsto {3 {b}^{2} = 24 \times 2}$

$\leadsto { {b}^{2} = \dfrac{48}{3}}$

$\leadsto { {b}^{2} = 16}$

$\leadsto {b = \sqrt{16}}$

$\leadsto {b = 4 \: m}$

So, the breath of the room = 4 m.

Therefore, length :-

$= \dfrac{3b}{2}$

$= \dfrac{3 \times 4}{2}$

$= 6 \: m$

So, the length of the room = 6 m.

Now , the rate of painting the walls is Rs. 5.5 per m^2.

And the total cost of painting is Rs. 1320.

So,

The area of all the 4 walls = $\dfrac{1320}{5.5}$

$\leadsto {2(length + breadth) \times height = \dfrac{1320}{5.5}}$

$\leadsto {2(6 + 4) \times h = 1320 \times \dfrac{10}{55}}$

$\leadsto {2(10) \times h = 240}$

$\leadsto {20 \times h = 240}$

$\leadsto {h = \dfrac{240}{20}}$

$\leadsto {h = 12 \: m}$

So, the height of the room = 12 m.

Therefore, the dimensions are :-

Height = 12 m

Breadth = 4 m

Length = 6 m.

_____________________