Question

Three equal cubes are placed adjacently in a row. Find the ratio of the total surface area of the resulting cuboid to the sum of the total surface areas of the three cubes.
(a) 7: 13
(b) 7: 3
(c) 7: 9
(d) 7: 5

Answer 1

Hint: Here, we need to find the ratio of the total surface area of the resulting cuboid to the sum of the total surface areas of the three cubes. First, we will find the sum of the surface area of the three cubes. Then, we need to find the dimensions of the resulting cuboid when the three cubes are joined end to end. Then, we will use the formula for total surface area of a cuboid and then, calculate the ratio of the total surface area of the resulting cuboid to the sum of the total surface areas of the three cubes.

Formula used:
We will use the following formulas:
1.The total surface area of a cube is given by the formula $$6a^2$$, where $$a$$ is the length of the side of the cube.
2.The total surface area of a cuboid is given by the formula $$2\left(lb+bh+lh\right)$$, where $$l$$ is the length, $$b$$ is the breadth, and $$h$$ is the height.

Complete step-by-step answer:
First, we will find the total surface areas of the three cubes.
It is given that the three cubes are equal.
Therefore, the volume, area, and edge of the cubes are the same.
Let the length of the side of the cubes be $$x$$.
The total surface area of a cube is given by the formula $$6a^2$$, where $$a$$ is the length of the side of the cube.
Substituting $$a=x$$ in the formula, we get
 Total surface area of 1 cube $$=6x^2$$
Thus, we get
Total surface area of the other two cubes $$=6x^2$$
Now, we will find the sum of the total surface areas of the three cubes.
The sum of the total surface areas of the three cubes $$=6x^2+6x^2+6x^2$$.
Adding the like terms of the expression, we get
The sum of the total surface areas of the three cubes $$=18x^2$$.
Now, let us draw the diagram to show how the resulting cuboid looks.

We can observe that when the three cubes are joined together end to end, the breadth and height of the resulting cuboid is equal to the side of the cubes, that is $$x$$.
The length of the cuboid is the sum of the lengths of the sides of the three cubes.
Therefore, the length of the cuboid $$=x+x+x$$.
Adding the like terms, we get
$$\Rightarrow$$ The length of the cuboid $$=3x$$.
Now, we will find the total surface area of the resulting cuboid.
The total surface area of a cuboid is given by the formula $$2\left(lb+bh+lh\right)$$, where $$l$$ is the length, $$b$$ is the breadth, and $$h$$ is the height.
Substituting $$l=3x$$, $$b=x$$, and $$h=x$$ in the formula, we get
Total surface area of the resulting cuboid $$=2\left(3x\times x+x\times x+3x\times x\right)$$
Multiplying the terms in the parentheses, we get
$$\Rightarrow$$ Total surface area of the resulting cuboid $$=2\left(3x^2+x^2+3x^2\right)$$
Adding the like terms in the parentheses, we get
$$\Rightarrow$$ Total surface area of the resulting cuboid $$=2\left(7x^2\right)$$
Multiplying 2 by 7, we get
$$\Rightarrow$$ Total surface area of the resulting cuboid $$=14x^2$$
Finally, we will find the ratio of the total surface area of the resulting cuboid to the sum of the total surface areas of the three cubes.
Dividing the total surface area of the cuboid by the sum of the total surface areas of the three cubes, we get
$$\Rightarrow$$ Total surface area of the cuboid $$÷$$sum of the total surface areas of the three cubes $$={\displaystyle \frac{14x^2}{18x^2}}$$
Simplifying the expression, we get
$$\Rightarrow$$ Total surface area of the cuboid $$÷$$sum of the total surface areas of the three cubes $$={\displaystyle \frac{7}{9}}$$
Therefore, the ratio of the total surface area of the resulting cuboid to the sum of the total surface areas of the three cubes is 7: 9.
Thus, the correct option is option (c).
Note: We need to remember that the sum of the surface areas of the three cubes will not be equal to the surface area of the resulting cuboid, unlike volume.
We added the like terms in the solution. Like terms are the terms whose variables and their exponents are the same. For example, $$100x,150x,240x,600x$$ all have the variable $$x$$ raised to the exponent 1. Terms which are not like, and have different variables, or different degrees of variables cannot be added together. For example, it is not possible to add $$4x^2$$ to $$-9y^2$$.