Question

Find the volume of a cuboid if its surface area is 208 sq. cm and the ratio of length, breadth and height is 2 : 3 : 4.

Answer 1

Hint: We will assume the ratio as 2x : 3x : 4x. We know that the surface area of a cuboid is $2\left( lb+bh+hl \right)$. So, from this, we will get a relation, $2\left( 2x.3x+3x.4x+4x.2x \right)=208$. On solving this, we will get a quadratic equation, $6{{x}^{2}}+12{{x}^{2}}+8{{x}^{2}}=104$. We will solve this quadratic equation to get the value of the length, breadth and height of the cuboid and then apply it in the formula for volume of the cuboid, which is $l\times b\times h$.

Complete step-by-step answer:
It is given in the question that the surface area of a cuboid is 208 sq. cm and the ratio of length, breadth and height is 2 : 3 : 4 and we have been asked to find the volume of the cuboid. Let us assume the length, breadth, and height of the cuboid to be 2x, 3x and 4x respectively. We know that the surface area of a cuboid is $2\left( lb+bh+hl \right)$. We have been given that the surface area of the cuboid is 208 sq. cm. So, we get,
$2\left( lb+bh+hl \right)=208\ldots \ldots \ldots \left( i \right)$
Now, we will substitute the values of l = 2x, b = 3x and h = 4x in equation (i). So, we get,
$\begin{align}
  & 2\left( 2x.3x+3x.4x+4x.2x \right)=208 \\
 & \Rightarrow 2x.3x+3x.4x+4x.2x=104 \\
 & \Rightarrow 6{{x}^{2}}+12{{x}^{2}}+8{{x}^{2}}=104 \\
 & \Rightarrow 26{{x}^{2}}=104 \\
 & \Rightarrow {{x}^{2}}=\dfrac{104}{26} \\
 & \Rightarrow {{x}^{2}}=4 \\
 & \Rightarrow x=\sqrt{4} \\
 & \Rightarrow x=\pm 2 \\
\end{align}$
Since, we cannot have negative values for length, breadth, and height, we will neglect -2. Therefore, we get x = 2. We know that the ratio of length, breadth and height are in the ratio of 2 : 3 : 4, so,
Length $\left( 2x \right)=2\times 2=4$, Breadth $\left( 3x \right)=3\times 2=6$ and Height $\left( 4x \right)=4\times 2=8$.

Now, we know that the volume of a cuboid is $l\times b\times h$. So, we have l = 4, b = 6 and h = 8. So, we get the volume of the cuboid as, $4\times 6\times 8=24\times 8=192c{{m}^{2}}$.
Therefore, the volume of the cuboid is $192c{{m}^{2}}$.

Note: Most of the students end up with calculation mistakes while finding the surface area of the cuboid. Also, there is a possibility that they take the length, breadth, and height of the cuboid as 2, 3, and 4 respectively. These numbers are just the ratios of the length, breadth, and height and not their actual measurements, which needs to be calculated.