Question

The lateral surface area and total surface area of a cuboid are 238 cm2 and
382 cm2. If the height of the cuboid is 7 cm, find the volume of the cuboid.​

Answer 1

What we have to do ?

Here we have given the lateral surface area of the Cuboid is 238 cm² and Total surface area of the Cuboid is 382 cm² respectively. We need to find the volume of cuboid for that first we have to find out the length and breadth of the cuboid.

Then, we will plug the measurement of height in order to have the length of the cuboid in terms of variable. After that, by using Substituting method.

Let's solve it now :

Formulae need to know :

To find the Total Surface Area, we will use the Formula :

${\boxed{\sf {TSA = 2( Lb + bh + hl )}}}$

Where,

• L → Length of the Cuboid
• B → Breadth of the Cuboid
• H → Height of the Cuboid

To find the Lateral surface area , we will use the formula :

${\boxed{\sf {LSA = 2( Length + Breadth ) \times height}}}$

To find the Volume, we will use the formula:

${\boxed{\sf {Volume = Length \times breadth \times height}}}$

Now, let's do it :

⠀$\underline{\bf{SoluTion :}}$ ⠀

Now , it is said that the Curved Surface area is 238 cm²

So,

$\dashrightarrow\:\:\sf 2( l + b ) \times h = 238$

$\dashrightarrow\:\:\sf 2( l + b ) \times 7 = 238$

$\dashrightarrow\:\:\sf 2( l + b ) = \dfrac{238}{7}$

$\dashrightarrow\:\:\sf 2( l + b ) = 34$

$\dashrightarrow\:\:\sf ( l + b ) = \dfrac{34}{2}$

$\dashrightarrow\:\:\sf l + b = 17$

⠀⠀$\underline{\bf{l = 17 - b}}$ ⠀ Eq.(i)

Then, Given Total surface area of cuboid is 382 cm²

$\dashrightarrow\:\:\sf TSA = 2( Lb + bh + hl )$

We know ,

$:\implies\sf 2 ( lb + bh + hl ) = 382$

$:\implies\sf ( lb + bh + hl ) = \dfrac{382}{2}$

$:\implies\sf lb + bh + hl = 191$

Putting the values respectively :

• $\longrightarrow$l = 17 - b
• $\longrightarrow$h = 7

$:\implies\sf (17 - b) b + b \times 7 + 7( 17 - b) = 191$

$:\implies\sf 17b - b^{2} + 7b + 119 - 7b = 191$

$:\implies\sf 17b - b^{2} = 191 - 119$

$:\implies\sf 17b - b^{2} = 72$

$:\implies\sf b^{2} - 17b + 72 = 0$

By Splitting middle term method :

$\dashrightarrow\:\:\sf b^{2} - (8 + 9)b + 72$

$\dashrightarrow\:\:\sf b^{2} - 8b - 9b + 72$

$\dashrightarrow\:\:\sf b (b - 8) -9 (b - 8)$

$\dashrightarrow\:\:\sf (b - 9) (b - 8)$

$\dashrightarrow\:\:\sf b - 9 = 0$

$\dashrightarrow\:\:\sf b = 9$

$\dashrightarrow\:\:\sf b - 8 = 0$

$\dashrightarrow\:\:\sf b = 8$

Now, two value of breadth is there , both are correct.

Substituting the value of ' b ' in equation (1) to get the measurement of length.

• $\longrightarrow$l = 17 - b
• $\longrightarrow$l = 17 - 8
• $\longrightarrow$l = 9

$\bigstar\:\underline{\textbf{As per the Question :}}$

Now, for the Volume of the Cuboid :

$\dashrightarrow\:\:\sf Volume = (Length \times breadth \times height)$

• $\longrightarrow$ Length = 9
• $\longrightarrow$ Breadth = 8
• $\longrightarrow$Height = 7

Putting in it , we get -

$\dashrightarrow\:\:\sf Volume = (9 \times 8 \times 7)$

$\dashrightarrow\:\:\sf Volume = (72 \times 7)$

$\dashrightarrow\:\:\sf Volume = 504 cm^{3}$

$:\implies \underline{ \boxed{\sf Volume = 504 \: cm^{3}}}$

Not sure about the answer ?

Lets verify to confirm -

For Verification :

$:\implies \bf C.S.A = 238 \: cm^{2}$

$:\implies \bf 2 ( l + b ) \times h = 238 \: cm^{2}$

$:\implies \bf 2 ( 8 + 9 ) \times 7 = 238 \: cm^{2}$

$:\implies \bf 2 \times 17 \times 7 = 238 \: cm^{2}$

$:\implies \bf 34 \times 7 = 238 \: cm^{2}$

$:\implies \bf 238 = 238$

$:\implies \bf LHS = RHS$

$\underline{\sf Verified ! }$

Answer 2

Answer:

$lateral \: surface \: area = 238 {cm}^{2} \\ \\ \implies \: 2(l + b) \times h = 238 {cm}^{2} \\ \\ \implies \: 2(lb + bh) = 238 {cm}^{2} - - - - - (i)\\ \\ total \: surface \: area = 382 {cm}^{2} \\ \\ \implies \: 2(lb + bh + hl) = 382 {cm}^{2} - - - - - (ii) \\ \\ \\ (ii) - (i) \\ \\ \longrightarrow \: \: 2(lb + bh + hl) - 2(lb + bh)= 382 {cm}^{2} - 238 {cm}^{2} \\ \\ \longrightarrow \: 2hl = 144 \\ \\ \longrightarrow \: hl = 72 \\ \\ 7 \times l = 72 \\ \\ l = \frac{72}{7}$