An Online Store Is Shipping A Standard Soccer Ball With A Diameter Of 22 Cm. The Box Used Is A Cube-shaped Cardboard Box With Side Lengths Of 25 Cm. Calculate The Amount Of Empty Space Left In The Box After Placing The Ball Inside And Round To The Nearest Whole Number.
This article delves into a practical geometry problem involving the calculation of empty space within a cube-shaped box after a standard soccer ball is placed inside. Understanding such spatial relationships is crucial not only in academic contexts but also in real-world applications like packaging and logistics. The problem we'll address involves a soccer ball with a diameter of 22 cm being shipped in a cube-shaped cardboard box with side lengths of 25 cm. Our primary goal is to determine the amount of empty space remaining in the box after the ball is placed inside, rounding the result to the nearest whole number. This problem combines the concepts of volume calculation for spheres and cubes, offering a comprehensive understanding of how three-dimensional shapes interact within a confined space.
Understanding the Problem
The initial step in solving this problem involves visualizing the scenario. Imagine a standard soccer ball, which is essentially a sphere, placed inside a cube-shaped box. The dimensions are as follows: the soccer ball has a diameter of 22 cm, meaning its radius is half of that, which is 11 cm. The box, on the other hand, is a cube with each side measuring 25 cm. To find the empty space, we need to calculate the volume of the box and the volume of the soccer ball separately. The empty space will then be the difference between the two volumes. This problem highlights the importance of understanding geometric shapes and their properties, as well as the ability to apply formulas to real-world situations. The accurate calculation of volumes is essential in various fields, from manufacturing and engineering to everyday tasks like packing and shipping. By breaking down the problem into smaller, manageable steps, we can systematically arrive at the solution.
Calculating the Volume of the Cube
The first geometric shape we need to consider is the cube. A cube is a three-dimensional shape with six square faces, all of which are identical. The volume of a cube is calculated by multiplying the length of one side by itself three times. In mathematical terms, if 's' represents the length of a side, the volume (V_cube) is given by the formula: V_cube = s³. In our specific scenario, the cube-shaped box has sides that are 25 cm long. Therefore, to find the volume of the box, we need to calculate 25 cm * 25 cm * 25 cm. This calculation will give us the total space available inside the box before the soccer ball is placed in it. Accurate calculation of the cube's volume is crucial as it forms the basis for determining the empty space left after the soccer ball is placed inside. The simplicity of the cube's volume formula makes it a fundamental concept in understanding three-dimensional geometry and spatial relationships. Mastering this calculation is essential for solving problems related to packaging, storage, and other practical applications where space utilization is a key factor.
Step-by-Step Calculation
To calculate the volume of the cube, we use the formula V_cube = s³, where s = 25 cm. Therefore, the volume of the cube is: V_cube = 25 cm * 25 cm * 25 cm. First, we multiply 25 by 25, which equals 625. Then, we multiply 625 by 25 again to get the final volume. 625 * 25 = 15625. Thus, the volume of the cube-shaped box is 15625 cubic centimeters (cm³). This value represents the total space within the box that can be occupied. It's a significant figure in our calculation as it will be compared to the volume of the soccer ball to determine the empty space. The meticulous step-by-step calculation ensures accuracy and avoids potential errors, which is crucial in mathematical problem-solving. Understanding the process of calculating the volume of a cube is not only essential for this specific problem but also for various other geometric and practical applications.
Calculating the Volume of the Soccer Ball (Sphere)
The second shape we need to consider is the soccer ball, which is essentially a sphere. A sphere is a perfectly round three-dimensional object in which every point on the surface is equidistant from the center. The volume of a sphere is calculated using the formula V_sphere = (4/3) * π * r³, where 'π' (pi) is a mathematical constant approximately equal to 3.14159, and 'r' is the radius of the sphere. In our problem, the soccer ball has a diameter of 22 cm, which means its radius is half of that, i.e., 11 cm. To find the volume of the soccer ball, we need to substitute the value of the radius into the formula and perform the calculation. This will give us the amount of space the soccer ball occupies inside the box. Accurate calculation of the sphere's volume is essential for determining the empty space left in the box. Understanding the formula and its application is a fundamental aspect of geometry and is used in various fields, including engineering, physics, and computer graphics.
Applying the Sphere Volume Formula
To calculate the volume of the soccer ball, we use the formula V_sphere = (4/3) * π * r³, where r = 11 cm and π ≈ 3.14159. First, we calculate r³, which is 11 cm * 11 cm * 11 cm = 1331 cm³. Next, we multiply this by π (3.14159), giving us 1331 cm³ * 3.14159 ≈ 4188.79 cm³. Then, we multiply this result by 4, which yields 4188.79 cm³ * 4 ≈ 16755.16 cm³. Finally, we divide this by 3 to complete the calculation: 16755.16 cm³ / 3 ≈ 5585.05 cm³. Therefore, the volume of the soccer ball is approximately 5585.05 cubic centimeters. This value represents the space occupied by the soccer ball inside the box and is crucial for determining the remaining empty space. The step-by-step application of the formula ensures accuracy in the calculation, highlighting the importance of precision in mathematical problem-solving.
Determining the Empty Space
Now that we have calculated the volume of both the cube-shaped box and the soccer ball, we can determine the empty space left in the box after the ball is placed inside. The empty space is simply the difference between the volume of the box and the volume of the soccer ball. This can be expressed as: Empty Space = Volume of Cube - Volume of Soccer Ball. We previously calculated the volume of the cube to be 15625 cm³ and the volume of the soccer ball to be approximately 5585.05 cm³. By subtracting the volume of the soccer ball from the volume of the cube, we will find the amount of space that is not occupied by the ball. This calculation is a practical application of spatial reasoning and volume calculation, demonstrating how mathematical concepts can be used to solve real-world problems related to packaging, storage, and logistics. Understanding how to determine empty space is essential for optimizing space utilization and minimizing waste in various industries.
Subtraction and Rounding
To find the empty space, we subtract the volume of the soccer ball from the volume of the cube: Empty Space = 15625 cm³ - 5585.05 cm³. This subtraction gives us 10039.95 cm³. The problem asks us to round the result to the nearest whole number. Since the decimal part is .95, which is greater than .5, we round up to the next whole number. Therefore, the empty space in the box after placing the soccer ball inside is approximately 10040 cubic centimeters. This final result provides a clear and practical answer to the problem, demonstrating the application of geometric calculations in a real-world scenario. The process of subtraction and rounding ensures that we arrive at a precise and easily understandable answer, highlighting the importance of mathematical accuracy and attention to detail in problem-solving.
Conclusion
In conclusion, by systematically calculating the volumes of the cube-shaped box and the soccer ball, and then finding the difference, we determined that there are approximately 10040 cubic centimeters of empty space left in the box after the soccer ball is placed inside. This problem demonstrates the practical application of geometric principles in everyday situations. Understanding how to calculate volumes and spatial relationships is essential in various fields, including packaging, logistics, engineering, and design. The ability to visualize three-dimensional shapes and apply the correct formulas is a valuable skill that enhances problem-solving capabilities. By breaking down complex problems into smaller, manageable steps, we can arrive at accurate solutions and gain a deeper understanding of the underlying mathematical concepts. This exercise not only reinforces our knowledge of geometry but also highlights the importance of mathematical reasoning in real-world contexts.