Question

343 smaller but identical cubes have been put together to form a larger cube. This larger cube is now painted on all 6 faces. How many of the smaller cubes have at least one faces painted? Select one: a. 218 b. 196 c. 152 d. 198

Answer 1

Given:

343 small cubes are placed together to form a big cube

6 of the faces of the big cube are painted

Find:

Total number of cubes with at least one face painted.

Find the number length of the big cube:

$\text{Volume} = 343 \text { small cubes}$

$\text {Length} = \sqrt[3]{343}$

$\text {Length} = 7 \text { small cubes}$

Find the number of cubes that are painted:

$\text {Top Face} = 7 \times 7$

$\text {Top Face} = 49 \text { small cubes}$

$\text {Bottom Face} = 7 \times 7$

$\text {Bottom Face} = 49 \text { small cubes}$

$\text {Front Face} = 5 \times 7$

$\text {Front Face} = 35 \text{ small cubes}$

$\text {Back Face} = 5 \times 7$

$\text {Back Face} = 35 \text{ small cubes}$

$\text {Left Face} = 5 \times 5$

$\text {Left Face} = 25 \text{ small cubes}$

$\text {Right Face} = 5 \times 5$

$\text {RIght Face} = 25 \text{ small cubes}$

$\text{Total cubes} = 49 + 49 + 35 + 35 + 25 + 25$

$\text{Total cubes} = 218 \text { small cubes}$

Answer: 218 of the small cubes have at least one face painted.

Answer 2

Given :

343 small cubes make a big cube with 6 faces painted

To find :

No. of small cubes with at least one face painted

Solution:

1.We need to find the no. of cube held each side  :

 The volume of the cube given in the question = 343 units

 We get each side length = $\sqrt[3]{343}$ = 7 units

 

2. We need to find no. of small cubes painted at least one side

  = No. of cubes at edge each face $\times$ 3 + 8

    (Vertices) + no. of cubes held between  

    vertex of each face $\times$ 6  

 = 5$\times$4$\times$3 + 8 + 5$\times$5$\times$6

 = 60 + 8 + 150

 = 218 cubes

*Note: The cubes at the edge have two sides painted.  

Hence, there are 218 small cubes painted at least one side.