A formula for the volume \(V\) of a sphere with radius \(r\) is \(V=\frac{4}{3} \pi r^{3} .\) If the radius of a spherical rubber ball is 1\(\frac{1}{4}\) inches, what is its volume to the nearest cubic inch? A. 5 A. 7 C. 8 D. 16 E. 65

Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, and solids. In this context, we are examining a sphere, a three-dimensional solid that is perfectly symmetrical around its center, with all points on its surface equidistant from that central point. This distance is known as the radius (r) of the sphere.

To visualize a sphere, think about a basketball or the Earth. Understanding the properties of a sphere is crucial because it assists us to solve real-world problems involving spherical objects, like calculating the amount of air required to inflate a beach ball or estimating the volume of planets.

The formula for the volume of a sphere \( V = \frac{4}{3} \pi r^3 \) is derived from integral calculus, but for our purposes, it's enough to remember that it requires the radius as the sole input. Geometric formulas such as this one allow us to quantify the space that an object occupies, which is essential in fields such as manufacturing, construction, and science.

Math problem solving is a systematic process that involves understanding the problem, devising a plan, carrying out that plan, and then looking back to see if the results make sense. The exercise given involved determining the volume of a sphere given a certain radius. This requires several math problem-solving skills, including converting mixed numbers, substituting values into a formula, and rounding off numbers.

In our scenario, converting the radius to a decimal number simplifies the calculations. By processing the given fraction into a decimal, we reduce the complexity of the arithmetic and make it easier to fit into the volume formula. Once the value is substituted into the formula, careful calculation leads to the result, which then must be sensibly rounded to meet the question's requirements — in this case, to the nearest whole number. This process emulates the daily use of math where preciseness and practical approximation often go hand in hand.

Volume calculation is a fundamental aspect of understanding three-dimensional shapes. For a sphere, the volume denotes the amount of space it occupies, and it's computed using the formula \( V = \frac{4}{3} \pi r^3 \) where 'r' is the radius. The key is to ensure that the radius measurement is correct because the radius value is cubed, meaning small errors can greatly affect the final volume result.

In practice, to calculate the volume of this spherical rubber ball, we obtained the radius in a decimal form and plugged it into the volume formula. After the calculation, we arrived at approximately 8.181 cubic inches. Due to the nature of the question, we had to round the answer to the nearest whole number, which gives us a practical, usable result of 8 cubic inches. This exercise demonstrates a practical application of volume calculation, which is crucial in various fields such as industrial design, packaging, and material requirements planning.