Determining the Diagonal of a New Cube Formed by Merging Three Cubes

Understanding the Diagonal of a New Cube Formed by Merging Three Smaller Cubes

In this article, we will explore the problem of determining the diagonal of a new cube formed by melting three smaller cubes with specified edge lengths. The process involves calculating the volume of the original cubes, finding the edge length of the new cube, and subsequently determining its diagonal. Let's dive into the detailed steps and calculations.

Merging Three Cubes into One New Cube

Consider three cubes with edge lengths of 5 cm, 1 cm, and 4 cm respectively. If these cubes are melted together, the resulting mass will form a single larger cube. To determine the edge length of this new cube, we first calculate the volume of each of the three smaller cubes:

Volume of cube with edge 5 cm (5^3 125 , text{cm}^3) Volume of cube with edge 1 cm (1^3 1 , text{cm}^3) Volume of cube with edge 4 cm (4^3 64 , text{cm}^3)

The combined volume of these three cubes is:

(125 1 64 190 , text{cm}^3)

The edge length (x) of the new cube can be found by taking the cube root of the total volume:

(x sqrt[3]{190} approx 5.748897079 , text{cm})

Now that we have the edge length of the new cube, we can determine its diagonal by using the formula for the diagonal of a cube, which is (x sqrt{3}):

Diagonal (5.748897079 times sqrt{3} approx 9.96 , text{cm})

Step-by-Step Calculation for a New Cube

Let's consider another scenario: three cubes with edge lengths of 3 cm, 4 cm, and 6 cm. The volumes of these cubes are:

Volume of cube with edge 3 cm (3^3 27 , text{cm}^3) Volume of cube with edge 4 cm (4^3 64 , text{cm}^3) Volume of cube with edge 6 cm (6^3 216 , text{cm}^3)

The combined volume of these three cubes is:

Total volume (27 64 216 307 , text{cm}^3)

The edge length of the new cube (X) can be found by taking the cube root of the combined volume:

(X sqrt[3]{307} approx 6.75 , text{cm})

The diagonal of this new cube is calculated as:

Diagonal ( sqrt{3} times 6.75 approx 11.65 , text{cm})

Detailed Solution and Final Calculation

A more detailed solution is provided:

Given the edge lengths 3 cm, 4 cm, and 6 cm, we calculate the volume of each cube. Total volume of the new cube (3^3 4^3 6^3 27 64 216 307 , text{cm}^3) Let the edge length of the new cube be (x). Since the volume of the new cube is equal to the combined volume of the original cubes, we have: (x^3 307) (x sqrt[3]{307} approx 6.75 , text{cm}) The diagonal of the new cube is calculated as ( sqrt{3} times x ). ( sqrt{3} times 6.75 approx 11.65 , text{cm})

Thus, the diagonal of the new cube is approximately 11.65 cm.

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