Question

Three equal cubes are placed adjacently in a row. Find the ratio of the total surface area of the new cuboid to that of the sum of the surface areas of three cubes.

Answer 1

Hint: We will first let the side of cube be of $a$ units. Then, find the total surface area of the cube. There are 3 cubes, then multiply the total surface area of the cube with 3. Next, determine the length, breadth, and height of the cuboid formed by arranging cubes. Then, find the total surface of the cuboid. Next, find the ratio of the total surface area of the new cuboid to that of the sum of the surface areas of three cubes.

Complete step by step Answer:

Let each side of the cube be $a$ units.
We will first find the surface area of one cube.
We know that the total surface area of the cube is $6{a^2}$
But, there are three cubes, then the total surface area of all the three cubes is the product of 3 and the total surface area of one cube.
$3\left( {6{a^2}} \right) = 18{a^2}$
Let the cubes be arranged together to form a cuboid.
Then, the length of the cuboid will be $3a$ units, the breadth will be $a$ units and the height will also be $a$ units.
We know that the formula of the total surface area of the cuboid is $2\left( {lb + bh + hl} \right)$, where, $l$ is the length of the cuboid, $b$ is the breadth of the cuboid and $h$ is the height of the cuboid.
Then, the total surface area of the cuboid formed is $2\left( {\left( {3a} \right)a + a\left( a \right) + \left( a \right)\left( {3a} \right)} \right) = 2\left( {3{a^2} + {a^2} + 3{a^2}} \right)$
Which is equal to $2\left( {7{a^2}} \right) = 14{a^2}$
We want to find the ratio of the total surface area of the new cuboid to that of the sum of the surface areas of three cubes.
Therefore, the ratio can be written as, $\dfrac{{14{a^2}}}{{18{a^2}}}$
Ratio is always written in simplest form. Hence, $\dfrac{{14{a^2}}}{{18{a^2}}} = \dfrac{7}{9}$ or 7:9
Hence, the required ratio is 7:9

Note: Total surface area of the cube is the area of the surfaces of a 3-dimensional shape. When cubes are arranged to form a cuboid, some of their faces get combined and the area of those faces is not counted. That is why, both the total surfaces are not equal.