Question

13. The dimensions of a rectangular box are in the ratio of 2 :3 : 4 and the difference between the cost of covering it with a cloth at the rate of rupees 10 and rupees 11 per metre square is rupees 1,300. Find the dimensions of the
box.​

Answer 1

$\Large {\underline { \sf \orange{Explication \: of \: Steps :}}}$

Let,

• Length = 2x

• Breadth = 3x

• Height = 4x

(All the dimensions have been assumed in m (metres).)

As per the given question,

$\bf \red { \dag }$ The difference between the cost of covering it with a cloth at the rate of Rs. 10 and Rs. 11 per m² is rupees 1,300.

As it is stated that it has to be covered with a cloth. So, we need to find the total surface area of the box.

According to the question,

$\longrightarrow \sf { Cost_{(Rs. \: 11)} \times T.S.A - Cost_{(Rs. \: 10)} \times T.S.A = 1300 }$

$\bf \red { \dag }$ T.S.A of the box :

→ T.S.A = 2 ( lb + lh + bh )

→ T.S.A = 2 [ (2x × 3x) + (2x × 4x) + (3x × 4x) ] m²

→ T.S.A = 2 ( 6x² + 8x² + 12x² ) m²

→ T.S.A = 2 (26x²) m²

→ T.S.A = 52x² m²

$\bf \red { \dag }$ Cost of covering it with cloth of Rs. 11 per m².

→ Cost = T.S.A × Rs. 11

→ Cost = Rs. ( 52x² × 11 )

→ Cost = Rs. 572x²

$\bf \red { \dag }$ Cost of covering it with cloth of Rs. 10 per m².

→ Cost = T.S.A × Rs. 11

→ Cost = Rs. ( 52x² × 10 )

→ Cost = Rs. 520x²

$\bf \red { \dag }$ Finding value of x :

$\longrightarrow \sf { Cost_{(Rs. \: 11)} \times T.S.A - Cost_{(Rs. \: 10)} \times T.S.A = 1300 }$

$\longrightarrow \sf { 572{x}^{2} - 520{x}^{2} = 1300 }$

$\longrightarrow \sf {52{x}^{2} = 1300 }$

$\longrightarrow \sf {{x}^{2} = \cancel{\dfrac{1300}{52}} }$

$\longrightarrow \sf {{x}^{2} = 25 }$

$\longrightarrow \sf {x = \sqrt{25} }$

$\longrightarrow \sf {x = 5}$

Now, calculating dimensions:

$\longmapsto \sf { Length = 2x \: m}$

$\longmapsto \sf { Length = 2(5) \: m}$

$\longrightarrow$ ❝ $\boxed{ \sf \orange { Length = 10 \: m }}$ ❞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longmapsto \sf { Breadth = 3x \: m}$

$\longmapsto \sf { Breadth = 3(5) \: m}$

$\longrightarrow$ ❝ $\boxed{ \sf \orange { Breadth = 15 \: m }}$ ❞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longmapsto \sf { Height = 4x \: m}$

$\longmapsto \sf { Height = 4(5) \: m}$

$\longrightarrow$ ❝ $\boxed{ \sf \orange { Height = 20 \: m }}$ ❞