Q1:

2 cubes each of volume $64$ cm$^3$ are joined end to end. Find the surface area of the resulting cuboid.

Given:

Volume of each cube ($V$) = $64 \text{ cm}^3$.

Two such cubes are joined end to end to form a cuboid.

To Find:

The total surface area of the resulting cuboid.

Step 1: Determine the side length of the individual cubes.

Let the side length of each cube be $a$ cm.

The formula for the volume of a cube is given by:

$V = a^3$

Substituting the given volume:

$64 = a^3$

To find $a$, we take the cube root of both sides:

$a = \sqrt[3]{64}$

$a = 4 \text{ cm}$ [Since $4 \times 4 \times 4 = 64$]

Step 2: Determine the dimensions of the resulting cuboid.

When two cubes are joined end to end, the length of the resulting cuboid increases, while the breadth and height remain the same as the side of the cube.

Length ($l$) = $a + a = 4 + 4 = 8 \text{ cm}$

Breadth ($b$) = $a = 4 \text{ cm}$

Height ($h$) = $a = 4 \text{ cm}$

Step 3: Calculate the total surface area of the cuboid.

The formula for the total surface area ($TSA$) of a cuboid is:

$TSA = 2(lb + bh + lh)$

Substitute the values $l = 8$, $b = 4$, and $h = 4$ into the formula:

$TSA = 2((8 \times 4) + (4 \times 4) + (8 \times 4))$

$TSA = 2(32 + 16 + 32)$ [Performing multiplication inside the parentheses]

$TSA = 2(80)$ [Summing the values inside the parentheses: $32 + 16 + 32 = 80$]

$TSA = 160 \text{ cm}^2$

Final Answer: The total surface area of the resulting cuboid is 160 cm$^2$.