Three solid iron cubes of edges 8 cm, 10 cm and 12 cm are melted together to make a new cube. It is observed that 496 cm³ of the melted material is lost due to improper handling. The area (in cm²) of the whole surface of the newly formed cube is:
Solving the Newly Formed Cube Surface Area Problem
This problem involves calculating the volume of multiple cubes, determining the volume of the final cube after accounting for material loss, finding the edge length of the new cube, and finally, calculating its total surface area.
Calculating Initial Volumes of the Cubes
First, we find the volume of each of the three initial iron cubes. The formula for the volume ($V$) of a cube with edge length ($s$) is $V = s^3$.
- Cube 1: Edge length $s_1 = 8$ cm. Volume $V_1 = 8^3 = 8 \times 8 \times 8 = 512$ cm³.
- Cube 2: Edge length $s_2 = 10$ cm. Volume $V_2 = 10^3 = 10 \times 10 \times 10 = 1000$ cm³.
- Cube 3: Edge length $s_3 = 12$ cm. Volume $V_3 = 12^3 = 12 \times 12 \times 12 = 1728$ cm³.
Finding the Volume of the New Cube
Next, we sum the volumes of the initial cubes to get the total volume of iron available before melting.
Total initial volume = $V_1 + V_2 + V_3 = 512 + 1000 + 1728 = 3240$ cm³.
The problem states that 496 cm³ of material was lost during the process. We subtract this loss from the total initial volume to find the volume of the material that forms the new cube.
Volume of the new cube ($V_{new}$) = Total initial volume - Material lost $V_{new} = 3240 - 496 = 2744$ cm³.
Determining the Edge Length of the New Cube
Let the edge length of the newly formed cube be $s_{new}$. The volume of this new cube is given by $V_{new} = s_{new}^3$. We need to find $s_{new}$.
$s_{new}^3 = 2744$ cm³
To find $s_{new}$, we calculate the cube root of 2744:
$s_{new} = \sqrt[3]{2744}$ cm
We can determine this by checking potential cube roots. Since $10^3 = 1000$ and $20^3 = 8000$, the edge length is between 10 and 20. The number 2744 ends in 4, suggesting the cube root might end in 4. Let's test 14:
$14^3 = 14 \times 14 \times 14 = 196 \times 14 = 2744$.
Therefore, the edge length of the new cube is $s_{new} = 14$ cm.
Calculating the Surface Area of the New Cube
Finally, we need to find the total surface area of the newly formed cube. The formula for the total surface area ($A$) of a cube with edge length ($s$) is $A = 6s^2$.
Using the edge length $s_{new} = 14$ cm:
Surface Area ($A_{new}$) = $6 \times s_{new}^2$ $A_{new} = 6 \times 14^2$ $A_{new} = 6 \times (14 \times 14)$ $A_{new} = 6 \times 196$ $A_{new} = 1176$ cm².
The area of the whole surface of the newly formed cube is 1176 cm².
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