Finding Cuboid Length from Cube Volume

The problem asks us to find the length of a cuboid formed by reshaping a cube, given the cube's volume and the ratio of the cuboid's dimensions.

Given Information

• Volume of the cube, $V_{cube} = 324$ cm$^3$.
• The dimensions of the cuboid (length : width : height) are in the ratio $3 : 2 : 2$.

Volume Conservation Principle

When a solid object is molded into a different shape, its volume remains the same. Therefore, the volume of the resulting cuboid is equal to the volume of the original cube.

Volume of the cuboid, $V_{cuboid} = V_{cube} = 324$ cm$^3$.

Cuboid Dimensions

Let the dimensions of the cuboid be represented in terms of a common factor, $x$, based on the given ratio:

• Length, $l = 3x$
• Width, $w = 2x$
• Height, $h = 2x$

Calculating the Length

The volume of a cuboid is calculated using the formula $V = l \times w \times h$. We can set up an equation using the known volume and the expressions for the dimensions:

$V_{cuboid} = (3x) \times (2x) \times (2x)$

$324 = 12x^3$

Now, solve for $x$:

$x^3 = \frac{324}{12}$

$x^3 = 27$

Taking the cube root of both sides:

$x = \sqrt[3]{27}$

$x = 3$

Finally, calculate the length of the cuboid using $l = 3x$:

$l = 3 \times 3$

$l = 9 \text{ cm}$

The length of the cuboid is 9 cm.