Question

Sports Find the amount of leather needed to cover an official major league baseball if its diameter is \(7.4\) centimeters.

Short answer

About 171.54 cm² of leather is needed to cover the baseball.

Step-by-step solution

Understand the Problem

We need to find the surface area of a sphere, because a baseball is shaped like a sphere. The surface area will tell us how much leather is needed to cover it.

Know the Formula for Surface Area

The formula for the surface area of a sphere is given by \(A = 4\pi r^2\), where \(r\) is the radius of the sphere.

Calculate the Radius

Since we are given the diameter of the baseball as \(7.4\) centimeters, we need to find the radius. The radius \(r\) is half of the diameter: \(r = \frac{7.4}{2} = 3.7\) centimeters.

Substitute the Radius into the Formula

Now, substitute the radius into the formula for surface area: \[A = 4\pi (3.7)^2\].

Calculate the Surface Area

First, calculate \(3.7^2 = 13.69\). Then multiply by \(4\pi\) to get the surface area \[A = 4 \times 3.14159 \times 13.69 \approx 171.535\] square centimeters.

Interpret the Result

The surface area of the baseball is approximately \(171.535\) square centimeters. This is the amount of leather needed to cover the baseball.

Key concepts

Surface Area

The surface area is a crucial concept in geometry, especially when dealing with three-dimensional objects like spheres. It represents the total area that the surface of an object occupies. For a sphere, this area is what you would get if you could "unwrap" the sphere and lay its surface flat. In practical terms, like in our baseball exercise, figuring out the surface area tells us how much material is required to cover the object completely. For spheres, the formula to determine the surface area is given by

• \(A = 4\pi r^2\)

where \(A\) is the surface area and \(r\) is the radius. The use of pi, a fundamental constant in mathematics, highlights its circular nature, as spheres are closely related to circles.

Sphere

A sphere is a perfectly symmetrical object in three dimensions, much like the shape of a baseball. Every point on the surface of a sphere is the same distance from its center—this constant distance is what we call the radius. Because of this symmetry, all the diameter and circumference lines of a sphere are equal no matter where you measure them around the sphere. Spheres are significant in various mathematical problems and real-world applications due to their unique geometric properties. In our example, understanding how a baseball reflects the characteristics of a sphere helps in calculating how much leather is needed to completely wrap around it without any gaps.

Radius Calculation

The radius is a critical component that we need to know to solve for the surface area of a sphere. It is half of the sphere’s diameter and is used in the formula for surface area calculation. In the baseball problem, the diameter is given as 7.4 centimeters. Therefore, finding the radius involves a simple step:

• \(r = \frac{d}{2}\)

Substituting the given diameter into the formula, we calculate:\(r = \frac{7.4}{2} = 3.7\) centimeters. This radius is then inserted into the surface area formula. Always remember that accurate calculation of the radius is essential because any error here will cascade through to your final result.

Mathematics

Mathematics is the language and tool we use to describe the world around us. It allows us to solve problems systematically using established formulas and processes. In geometry, concepts like surface area require us to apply algebraic manipulation and arithmetic operations. These skills are crucial not just for academic exercises, but for real-world applications where precise measurements are necessary—think engineering, architecture, or even art. In our example with the baseball, mathematics enables the conversion of physical measurements into actionable data—like the amount of leather required—through clear and logical steps.

Problem-Solving

Problem-solving is a key skill honed through mathematics. It involves breaking down a problem into manageable parts and applying known principles to reach a solution. Each problem presents unique challenges, but there is often a structured method to tackle them. In the baseball problem, the steps involve:

• Identifying the shape (sphere) and what needs to be calculated (surface area).
• Understanding and applying the relevant formula.
• Substituting known values (diameter) to find unknowns (radius).
• Performing calculations to determine the final result.

Problem-solving combines logical reasoning with practical mathematics, making complex tasks achievable through methodical planning and execution.