Solving for the Dimensions of a Cuboid Given Its Volume and Ratio

In this article, we'll explore a step-by-step method to determine the dimensions of a cuboid when given its volume and the ratio of its sides. The example provided will illustrate how to find the actual dimensions based on a given volume and a specified ratio. This information is particularly useful for students and professionals involved in geometry, engineering, and architectural design.

Understanding the Problem

We are given that the dimensions of a cuboid are in the ratio 3:2:5, and the volume of the cuboid is 240 cubic centimeters (cc). Our goal is to determine the exact dimensions of the cuboid.

Step-by-Step Solution

The process involves defining the dimensions based on the given ratio and then using the volume formula for a cuboid to solve for the common multiplier.

Defining the Dimensions

To start, let the dimensions of the cuboid be defined as follows:

Length: (3x) Width: (2x) Height: (5x)

Here, (x) is a common multiplier. This approach simplifies the problem by expressing each dimension in terms of (x).

Applying the Volume Formula

The formula for the volume (V) of a cuboid is:

[V text{length} times text{width} times text{height}]

Substituting the defined dimensions into the formula, we get:

[V 3x times 2x times 5x]

Let's simplify this expression:

[V 3 times 2 times 5 times x^3 3^3]

Setting Up the Equation

We know the volume (V) is 240 cubic centimeters. Substituting this into the equation, we have:

[3^3 240]

Solving for (x^3)

To find (x^3), we divide both sides of the equation by 30:

[x^3 frac{240}{30} 8]

Calculating (x)

To solve for (x), we take the cube root of both sides of the equation:

[x sqrt[3]{8} 2]

Calculating the Dimensions

Now that we have (x 2), we can calculate the actual dimensions of the cuboid:

Length: (3x 3 times 2 6 text{ cm}) Width: (2x 2 times 2 4 text{ cm}) Height: (5x 5 times 2 10 text{ cm})

Therefore, the dimensions of the cuboid are 6 cm by 4 cm by 10 cm.

Conclusion

This method provides a clear and systematic approach to solving for the dimensions of a cuboid given its volume and the ratio of its sides. By following these steps, you can easily determine the dimensions and apply this knowledge in various real-world scenarios, such as in geometry, engineering, and architectural design.

Additional Resources

For further exploration and practice, consider the following resources:

Interactive Geometry Tools Cuboid Volume Calculator Practice Problems and Solutions