Question

The dimensions of a cuboid are in the ratio 5:2:1. Its volume is 1250 cubic meters. find the total surface area.

Answer 1

Hint: since the volume of a cuboid is given and the dimensions of a cuboid, that is length, breadth and height of a cuboid is given in terms of ratio, we can solve it by using the concept of ratios. Thereby we can find the total surface area of a cuboid using the formula \[{\text{TSA of a cuboid = 2(lb + bh + hl)}}\].

Complete step by step answer:
Given volume \[(v)\]\[ = 1250\]
let the dimensions of the cuboid be
Length \[(l) = 5x\]
Breadth \[(b) = 2x\]
Height \[(h) = 1x\]
We can calculate the required dimensions by using the given data in the problem. Since volume of a cuboid is given therefore, we have
Volume \[(v) = l \times b \times h\]
Substituting the values, we get
\[1250 = (5x) \times (2x) \times (1x)\]
\[ \Rightarrow 1250 = 10{x^3}\]
\[ \Rightarrow 125 = {x^3}\]
\[ \Rightarrow x = {\left( {125} \right)^{\dfrac{1}{3}}}\]
Therefore, \[x = 5\]
Therefore, substituting the value of x we can obtain the dimensions and the dimensions are given by:
Length \[(l) = 5x = 5 \times 5 = 25m\]
Breadth \[(b) = 2x = 2 \times 5 = 10m\]
Height \[(h) = 1x = 1 \times 5 = 5m\]
Now the formula to find total surface area of a cuboid is given by,
\[{\text{TSA = 2(lb + bh + hl)}}\]
Now substituting the values of l, b, h in the above formula we get
\[{\text{TSA = 2(25}} \times {\text{10 + 10}} \times 5{\text{ + 5}} \times 25{\text{)}}\]
On simplification we get
\[ \Rightarrow TSA = 2(250 + 50 + 125)\]
\[ \Rightarrow TSA = 2(425)\]
\[ \Rightarrow TSA = 850{m^2}\]
Thus, the required total surface area of the cuboid is \[850{m^2}\].

Note: In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. The main difference between a cube and a cuboid is that a cube has six square-shaped faces of the same size but a cuboid has rectangular faces. A cuboid is rectangular-shaped and a cube is square-shaped.