Question

There are six faces in a cube, Rajeev fix one cube on each of the faces. The dimensions of all the cubes are same. What is the ratio of total surface area of the newly formed solid to the area of a single cube? (a) \(7: 1\) (b) \(6: 1\) (c) \(5: 1\) (d) \(41: 9\)

Short answer

**Short Answer:** The ratio of the total surface area of the newly formed solid to the area of a single cube is \(11:2\). However, none of the available options match this answer. It is important to notify the student or teacher about this discrepancy.

Step-by-step solution

Find the surface area of the central cube

The central cube has six faces, but since new cubes are attached to each face, only a portion of each face remains exposed. The exposed area of each face of the central cube is equal to half of the area of each face of the small cubes (since every exposed face is half the width on each side). So, the total exposed surface area of the central cube is \(6(\frac{1}{2}x^2) = 3x^2\), where x is the side of the small cubes.

Find the surface area of the attached cubes

There are six attached cubes, and each cube has one face fully covered by the central cube, leaving five exposed faces. The surface area of a cube is given by the formula \(6x^2\), so the surface area of the exposed faces of an attached cube is \(5x^2\). Since there are six attached cubes, the total surface area for the attached cubes is \(6(5x^2) = 30x^2\).

Find the total surface area of the newly formed solid

To find the total surface area of the newly formed solid, we add the exposed surface area of the central cube and the surface area of the attached cubes. Therefore, the total surface area is \(3x^2 + 30x^2 = 33x^2\).

Find the surface area of a single cube

The surface area of a cube is given by the formula \(6x^2\), so the surface area of a single attached cube is \(6x^2\).

Find the ratio of the total surface area to the area of a single cube

To find the ratio of the total surface area of the newly formed solid to the area of a single cube, we can divide the total surface area by the surface area of a single cube: \(\frac{33x^2}{6x^2} = \frac{33}{6} = \frac{11}{2}\). So, the ratio of the total surface area of the newly formed solid to the area of a single cube is \(11:2\). However, none of the available options match the solution we have just found. It is important to notify the student or teacher about this discrepancy.

Key concepts

Geometric Solids

When we talk about geometric solids, we're referring to three-dimensional objects such as cubes, spheres, cones, and cylinders. These shapes exist in the real world and have volume, unlike two-dimensional shapes that only have area. A cube, one of the simplest and most symmetrical geometric solids, is a perfect example to explore.

A cube has six identical square faces, twelve equal edges, and eight vertices where the edges meet. The uniformity of its dimensions makes it an excellent candidate for introducing the concepts of surface area and volume. Understanding the properties of a cube is crucial as it helps students to comprehend more complex solids later in their studies.

Visualizing geometric solids can sometimes be challenging, so it's helpful to have manipulatives or 3D models to explore their properties hands-on. When students have a solid grasp of basic geometric solids like cubes, they are better prepared to tackle more complicated shapes and their calculations.

Ratio and Proportion

Ratio and proportion are fundamental concepts in mathematics that relate two quantities to each other. A ratio represents the relationship between two or more numbers, showing how many times one value is contained within another. When two ratios are equal, they are said to be in proportion.

In the context of our cube problem, the ratio is used to compare the surface areas of the newly formed solid (with cubes fixed on each face of a central cube) to the surface area of a single cube. Understanding how to calculate and simplify ratios is an essential skill when analyzing the relative sizes of two quantities, especially in problems involving surface areas or volumes.

For example, if the surface area of the larger structure is twice that of a single cube, the ratio would be 2:1. Ratios can also provide a clearer understanding of scale in real-world problems, helping students to conceptualize things such as maps or models.

3D Shapes Surface Area Calculation

Calculating the surface area of 3D shapes is a common problem in geometry. The surface area is the total area that the surface of an object occupies. Each shape has its formula based on its unique attributes. For our cube exercise, the formula to find the surface area of a single cube is \(6x^2\), where 'x' is the length of one side.

An interesting aspect of this calculation is considering what happens to the surface area when other cubes are attached to each face of a central cube. Not every face is entirely visible or exposed, and therefore, the calculations must adjust to recognize the partial exposures and overlaps. This particular exercise highlights the operational aspect of surface area calculations that deals with adding or subtracting areas as the shape changes.

Students must understand that when working with composite 3D structures, they might need to apply the surface area formulas multiple times and combine the results to find the total surface area. Being adept at breaking down complex shapes into simpler components for calculation is a valuable skill and demonstrates a deeper comprehension of 3D geometry.