Question

The length, breadth and height of a cuboid are in the ratio 4: 3:2.
(i) If its surface area is 3328 cm?,
find its volume (ii) If its volume is 5184 cm3, find its surface area.
​

Answer 1

S O L U T I O N :

We have length, breadth & height of a cuboid are in the ratio 4:3:2 .

1. Their surface area is 3328 cm² .
2. Their it's volume is 5184 cm³ .

$\underline{\bf{Explanation\::}}$

$\underbrace{\bf{1^{st}\:condition\::}}$

Let the ratio of the cuboid be r .

As we know that formula of the total surface area of a cuboid;

$\boxed{\bf{2(lb+bh+lh)\:\:\:[sq.unit]}}$

We have;

• Length, (l) = 4r
• Breadth, (b) = 3r
• Height, (h) = 2r

A/q

$\mapsto\tt{2[(4r\times 3r) +(3r\times 2r)+(4r\times 2r)]=3328}$

$\mapsto\tt{2[(12r^{2} ) +(6r^{2})+(8r^{2})]=3328}$

$\mapsto\tt{2[26r^{2}]=3328}$

$\mapsto\tt{52r^{2}=3328}$

$\mapsto\tt{r^{2}=\cancel{3328/52}}$

$\mapsto\tt{r^{2} = 64}$

$\mapsto\tt{r = \sqrt{64} }$

$\mapsto\bf{r = 8\:cm}$

Now;

$\bullet\sf{Length \:of\:cuboid=4r = (4\times 8)cm = \boxed{\bf{32\:cm}}}$

$\bullet\sf{Breadth \:of\:cuboid=4r = (3\times 8)cm = \boxed{\bf{24\:cm}}}$

$\bullet\sf{Height \:of\:cuboid=2r = (2\times 8)cm = \boxed{\bf{16\:cm}}}$

As we know that formula of the volume of cuboid;

$\boxed{\bf{Volume\:of\:cuboid=lbh\:\:\:(cubic\:unit)}}$

$\mapsto\tt{Volume\:of\:cuboid=Length \times Breadth \times Height}$

$\mapsto\tt{Volume\:of\:cuboid=32 \times 24 \times 16}$

$\mapsto\bf{Volume\:of\:cuboid=12288\:cm^{3}}$

$\underbrace{\bf{2^{nd}\:condition\::}}$

$\mapsto\sf{Volume \:of\:cuboid=length\times breadth \times height }$

$\mapsto\sf{5184=4r\times 3r \times 2r}$

$\mapsto\sf{5184=24r^{3}}$

$\mapsto\sf{\cancel{5184/24}=r^{3}}$

$\mapsto\sf{216=r^{3}}$

$\mapsto\sf{3\sqrt{216} =r}$

$\mapsto\bf{r=6\:cm}$

Now;

$\bullet\sf{Length \:of\:cuboid=4r = (4\times 6)cm = \boxed{\bf{24\:cm}}}$

$\bullet\sf{Breadth \:of\:cuboid=3r = (3\times 6)cm = \boxed{\bf{18\:cm}}}$

$\bullet\sf{Height \:of\:cuboid=2r = (2\times 6)cm = \boxed{\bf{12\:cm}}}$

$\boxed{\bf{2(lb+bh+lh)\:\:\:[sq.unit]}}$

→ TSA of Cuboid = 2[(24×18) + (18×12) + (24×12)] cm²

→ TSA of Cuboid = 2[432 + 216 + 288] cm²

→ TSA of Cuboid = 2[936] cm²

→ TSA of Cuboid = 2 × 936 cm²

→ TSA of Cuboid = 1872 cm²

Answer 2

Answer:

1872 cm³ according to the question