Question

What is the total surface area of the identical cubes of largest possible size that are cut from a cuboid of size \(85 \mathrm{~cm} \times 17 \mathrm{~cm} \times 5.1 \mathrm{~cm} ?\) (a) \(26010 \mathrm{~cm}^{2}\) (b) \(21600 \mathrm{~cm}^{2}\) (c) \(26100 \mathrm{~cm}^{2}\) (d) none of these

Short answer

Answer: The total surface area of the identical cubes cut from the given cuboid is \(1734 \mathrm{~cm}^{2}\).

Step-by-step solution

Determine the largest possible side length of the cubes

We need to find the greatest common divisor (GCD) of the given dimensions of the cuboid, \((85, 17, 5.1)\). In this case, since 85 and 17 are both integers and 5.1 is a decimal, we first need to convert 5.1 to a fraction. 5.1 is equal to \(\frac{51}{10}\), so now we have (85, 17, \(\frac{51}{10}\)) as dimensions. The GCD of the numerators (85, 17, 51) is 1, and the LCM (least common multiple) of the denominators (1, 1, 10) is 10. Hence, the largest possible side length of the cubes is \(\frac{1}{10}\) cm. This value, however, is not reasonable for our problem, as it would imply that the side length of each smaller cube is much smaller than the given dimensions of the cuboid. In this case, we consider the greatest integer divisor common to the dimensions (85, 17), which is 17. So, the largest possible side length of the cubes is 17 cm.

Calculate the number of cubes

To find the number of cubes, we need to divide the volume of the cuboid by the volume of a single cube: Number of cubes \(= \frac{(85 \times 17 \times 5.1)}{(17 \times 17 \times 17)}\) Simplify the fraction: Number of cubes \(= \frac{5.1}{17} \approx 0.3\) Since we need to have a whole number of cubes, this means we can make 1 cube of length 17 cm.

Calculate the surface area of a single cube

The surface area \(A\) of a cube with side length \(s\) can be calculated using the formula: \(A = 6s^2\) Here, \(s = 17 \mathrm{~cm}\). So, the surface area of one cube is: \(A = 6 \times (17)^{2} = 6 \times 289 = 1734 \mathrm{~cm}^{2}\)

Calculate the total surface area of all the cubes

Now, we need to multiply the surface area of one cube by the total number of cubes to find the total surface area: Total surface area \(= 1734 \times 1\) Total surface area \(= 1734 \mathrm{~cm}^{2}\) As we can see, none of the options given in the exercise corresponds to our calculated total surface area: 1743 ≠ (a) 26010 1743 ≠ (b) 21600 1743 ≠ (c) 26100 So, the correct answer is: (d) none of these

Key concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is an important concept in mathematics that helps us find the largest number that divides two or more numbers without leaving a remainder. When dealing with dimensions of a cuboid, finding the GCD can help us determine the largest possible integer length that can be the side of small, identical cubes cut from this cuboid.
To find the GCD, we look at all the lengths given. In this case, we have 85, 17, and 5.1 cm. The process involves converting any decimal dimensions into a more workable fraction or integer form. For the dimension 5.1 cm, it is converted to a fraction as \(\frac{51}{10}\).
Next, we find the GCD of the numerators. Here, the numerators are 85, 17, and 51. The GCD of these numbers is 1, indicating no common factor greater than one exists between them.
Due to the presence of 5.1 in decimals, the realistic approach is to check the integer dimensions, simplifying to the GCD of 85 and 17, which is 17 cm.
This approach allows us to define the largest possible cube size that can be feasibly cut from the cuboid.

Volume Calculation

Volume calculation is a fundamental concept that involves finding the amount of space occupied by a three-dimensional object. The volume of a cuboid is calculated by multiplying its length, width, and height together. For a given cuboid with dimensions 85 cm, 17 cm, and 5.1 cm, the volume is computed as follows:\[\text{Volume of Cuboid} = 85 \times 17 \times 5.1\]This resulting product will give the volume in cubic centimeters.When calculating the number of identical cubes that can be created from the cuboid, the volume of each cube needs to be determined. A cube with a side length of 17 cm has a volume:\[\text{Volume of Cube} = 17 \times 17 \times 17\]This simplifies to a more manageable number, allowing you to check how many such cubes fit into the cuboid:\[\text{Number of Cubes} = \frac{85 \times 17 \times 5.1}{17 \times 17 \times 17}\]This division helps determine the number of cubes but, in this scenario, only allows for the creation of one whole cube of the specified dimension.

Cuboid Dimensions

Understanding the dimensions of a cuboid is crucial when dealing with problems involving cutting out identical cubes. A cuboid is a three-dimensional figure with six rectangular faces, and its dimensions can determine the size of cubes that fit in it.
The given problem deals with a cuboid having sides of 85 cm, 17 cm, and 5.1 cm. Although each side can individually be seen as a potential side for a cube, the practical approach is to focus on the greatest integer dimension, ensuring each dimension can accommodate cube sides evenly.
In this case, the practical integer value is 17 cm since a cube side of this length can evenly fit within both the 85 cm and 17 cm sides of the cuboid. This ensures that cubes cut from the cuboid use as much material with minimal waste.

• Identifying the largest possible integer length helps in maximizing cube size and minimizing leftover materials.
• Evaluating dimensions in terms of practical and theoretical is vital to align with the physical constraints of the problem.

This understanding aids in determining possible configurations and potential optimization of material use.