Question

What is the volume of a cuboid having surface area 208 cm² and the sides in
the ratio 3: 4:2?
​

Answer 1

Answer:

• Volume of the cuboid is 192 cm³.

Step-by-step explanation:

Given that:

• Total surface area of a cuboid is 208 cm².
• Sides of the cuboid are in the ratio 3 : 4 : 2.

To Find:

• Volume of the cuboid.

Formulas Used:

Total surface area of a cuboid:

• 2(LB + BH + LH)

Volume of a cuboid:

• (L × B × H)

Where,

• L = Length of the cuboid
• B = Breadth of the cuboid
• H = Height of the cuboid

Let us assume:

• Length of the cuboid = 4y
• Breadth of the cuboid = 3y
• Height of the cuboid = 2y

As we know that:

Total surface area of a cuboid = 2(LB + BH + LH) sq. units

Substituting the values,

$\sf{208 = 2\{(4y\times 3y)+(3y\times 2y)+(4y\times 2y)\}}$

Opening the brackets,

$\sf{208=2\{12y^2+6y^2+8y^2\}}$

Adding 12y², 6y² and 8y²,

$\sf{208 = 2\times26y^2}$

Multiplying 2 and 26y²,

$\sf{208 = 52y^2}$

Transposing 52 from RHS to LHS and changing its sign,

$\sf{\dfrac{208}{52}=y^2}$

Dividing 208 by 52,

$\sf{4 = y^2}$

Solving further,

$\sf{\sqrt{4}= y}\\\\\sf{y = \pm 2}$

(Since length is never negative. So here we have to neglect the negative value of of y.)

Thus, y = 2

Therefore,

• Length of cuboid = 4y = 4 × 2 = 8 cm
• Breadth of cuboid = 3y = 3 × 2 = 6 cm
• Height of cuboid = 2y = 2 × 2 = 4 cm

Now,

Volume of the cuboid = (L × B × H)

= (8 × 6 × 4) cm³

= 192 cm³

Hence, volume of the cuboid is 192 cm³.