Question 14 Marks
Interlocking cubes come in different colours. Different shapes can be created by joining them.
Sasha is making large cube using small interlocking cubes. She starts with a yellow cube and then fits one layer of red cubes around it to make the large cube.
On the basis of above given information, answer the following questions.
(i) What is the number of red cubes used?
(a) 9 $~~\quad$ (b) 18
(c) 26 $\quad$ (d) 27
(ii) Sasha considers the red layer as the first layer on the yellow cube. Which equation can be used to find the number of red cubes?
(a) $6 x^2$ $~\qquad$ (b) $x^3$
(c) $x^3+1$ $\quad$ (d) $2(x+2)^2+4 x^2$
(iii) The red cube is surrounded by a layer of green cubes. The resulting figure is also a cube. What is the number of green cubes used?
(iv) Shubham puts $\text x$ layers around the yellow cube. How many small cubes did he use on the top-most face of the large cube so formed?
Sasha is making large cube using small interlocking cubes. She starts with a yellow cube and then fits one layer of red cubes around it to make the large cube.
On the basis of above given information, answer the following questions.
(i) What is the number of red cubes used?
(a) 9 $~~\quad$ (b) 18
(c) 26 $\quad$ (d) 27
(ii) Sasha considers the red layer as the first layer on the yellow cube. Which equation can be used to find the number of red cubes?
(a) $6 x^2$ $~\qquad$ (b) $x^3$
(c) $x^3+1$ $\quad$ (d) $2(x+2)^2+4 x^2$
(iii) The red cube is surrounded by a layer of green cubes. The resulting figure is also a cube. What is the number of green cubes used?
(iv) Shubham puts $\text x$ layers around the yellow cube. How many small cubes did he use on the top-most face of the large cube so formed?
Answer
View full question & answer→(i) (c) To make first cube,
Number of required cube $=1$
To make second cube,
Number of required cube $=3^3=27$
Since, large cube is made by interlocking red cubes around the yellow cube placed at centre.
$\therefore$ Number of red cubes used $=27-1=26$
(ii) (d) By hit and trial method, if we put $x=1$ in given equations,
$2(x+2)^2+4 x^2+4 x=2(1+2)^2+4(1)^2+4(1)=26$ which is the number of red cubes used.
Therefore, $2(x+2)^2+4 x^2+4 x$ can be used to find the number of red cubes.
(iii) Now, the red cube is surrounded by green cubes.
Since, we know the number of cubes required to make first cube $=1$
The number of cubes required to make second cube $=3^3=27$
$\therefore$ Number of cubes required to make third cube
$=5^3=125$
$\therefore$ Number of green cubes used $=125-27=98$
(iv) When Shubham puts red cubes around the yellow cube.
The number of small cubes on the top-most face of cube $=9$
When green cubes are put around the yellow cube.
The number of small cubes on the top-most face of cube $=25$
If we keep on repeating the process, the numbers we will get on the top are
$1,9,25,49,81, \ldots \ldots \ldots$
Since, Shubham puts x layers
$\therefore$ The equation to find small cube on the top-most face can be given by $(2 x+1)^2$, where $x$ is whole number.
(v) 2 cubes and 3 cubes.s
Number of required cube $=1$
To make second cube,
Number of required cube $=3^3=27$
Since, large cube is made by interlocking red cubes around the yellow cube placed at centre.
$\therefore$ Number of red cubes used $=27-1=26$
(ii) (d) By hit and trial method, if we put $x=1$ in given equations,
$2(x+2)^2+4 x^2+4 x=2(1+2)^2+4(1)^2+4(1)=26$ which is the number of red cubes used.
Therefore, $2(x+2)^2+4 x^2+4 x$ can be used to find the number of red cubes.
(iii) Now, the red cube is surrounded by green cubes.
Since, we know the number of cubes required to make first cube $=1$
The number of cubes required to make second cube $=3^3=27$
$\therefore$ Number of cubes required to make third cube
$=5^3=125$
$\therefore$ Number of green cubes used $=125-27=98$
(iv) When Shubham puts red cubes around the yellow cube.
The number of small cubes on the top-most face of cube $=9$
When green cubes are put around the yellow cube.
The number of small cubes on the top-most face of cube $=25$
If we keep on repeating the process, the numbers we will get on the top are
$1,9,25,49,81, \ldots \ldots \ldots$
Since, Shubham puts x layers
$\therefore$ The equation to find small cube on the top-most face can be given by $(2 x+1)^2$, where $x$ is whole number.
(v) 2 cubes and 3 cubes.s