Question

The lateral surface area of a cuboid is \[224{\text{ c}}{{\text{m}}^2}\]. Its height is \[7{\text{ cm}}\] and the base is square. Find
(a) side of the square base
(b) the volume of the cuboid

Answer 1

Hint: Here, we have the lateral surface area given in the question and also the height is given. We will use the formula for lateral surface area and equate it to the given area. Also, since the base of the cuboid is a square therefore its length will be equal to its breadth because all the sides of a square are equal in length.

Formula used:
Lateral surface area of a cuboid \[ = 2\left( {l + b} \right) \times h\]
Volume of cuboid \[ = l \times b \times h\]
where, \[l = {\text{ length}}\], \[b = {\text{ breadth}}\] and \[h = {\text{ height}}\].

Complete step by step answer:

(a) Given: Lateral surface area of cuboid \[ = 224{\text{ c}}{{\text{m}}^2}\]
Height \[ = 7{\text{ cm}}\]
Also, we know that the lateral surface area of cuboid is \[ = 2\left( {l + b} \right) \times h\]
So, we will equate it to the given lateral surface area.
\[\therefore 2\left( {l + b} \right) \times h = 224{\text{ c}}{{\text{m}}^2}\]
Now we will put the value of height in the above equation. So, it becomes,
\[ \Rightarrow 2\left( {l + b} \right) \times 7{\text{ cm}} = 224{\text{ c}}{{\text{m}}^2}\]
Now we will shift \[7\] and \[2\] to the denominator of the right-hand side. So, we get,
\[ \Rightarrow \left( {l + b} \right) = \dfrac{{224{\text{ c}}{{\text{m}}^2}}}{{2 \times 7{\text{ cm}}}}\]

On further simplification we get,
\[ \Rightarrow \left( {l + b} \right) = 16{\text{ cm}}\]
Now, it is given in the question that the base of a cuboid is a square. So, its length and breadth will be equal.
\[l = b = {\text{ side of square}}\]
\[ \Rightarrow 2 \times side = 16{\text{ cm}}\]
\[ \Rightarrow side = \dfrac{{16}}{2}{\text{ cm}}\]
On dividing we get,
\[ \therefore side = 8{\text{ cm}}\]

Hence, the side of the square base is \[8{\text{ cm}}\].

(b) Now we have
\[l = b = 8{\text{ cm}}\] and \[h = 7{\text{ cm}}\]
We know that the volume of cuboid \[ = l \times b \times h\]
So, we will put the values of length breadth and height in the above formula to get the volume.
\[{\text{ volume of cuboid}} = l \times b \times h\]
\[\Rightarrow {\text{ volume of cuboid}} = \left( {8 \times 8 \times 7} \right){\text{ c}}{{\text{m}}^3}\]
\[\therefore {\text{ volume of cuboid}} = 448{\text{ c}}{{\text{m}}^3}\]

Hence, the volume of the cuboid is $448{\text{ c}}{{\text{m}}^3}$.

Note: In this question all the data given are in the same units so we haven’t changed any one of them. But in some other questions they might be in different units. So, there we need to change them and get all of them into the same units. Another point to note is that lateral surface area is not the same as total surface area.