91Ó°ÊÓ

Geometry is the area of mathematics that studies the sizes, shapes, and properties of figures and spaces. It's fundamental to many areas of science and engineering. In the context of this exercise, geometry helps us understand the properties of a sphere and how to calculate its surface area. A sphere is a perfectly round 3D shape, like a ball. All points on a sphere's surface are equidistant from its center. Understanding these basic properties is crucial for using formulas to find measurements such as surface area.

The surface area of a sphere is the total area covered by its surface. To find this, we use a specific formula. The formula for the surface area of a sphere is: \[A = 4 \pi r^2\] Here, \(A\) represents the surface area, \(r\) is the radius of the sphere, and \(\pi\) (pi) is a constant approximately equal to 3.1416. Radius is the distance from the center of the sphere to any point on its surface. This formula tells us that the surface area depends on the square of the radius: when the radius increases, the surface area increases much faster. This relationship is key to understanding how changes in size affect the surface area.

Let's apply this information to our specific problem. We're given a sphere with a radius of 0.75 feet.
Here's the step-by-step process:
1. Start with the formula for surface area: \(A = 4 \pi r^2\)
2. Substitute the given radius into the formula: \[A = 4 \pi (0.75)^2\]
3. Calculate \((0.75)^2\): \(0.75 \times 0.75 = 0.5625\)
4. Multiply this by 4 and \(\pi\): \(A = 4 \pi \times 0.5625\)
5. Finally, multiply the values to get the approximate surface area: \(4 \times 3.1416 \times 0.5625 = 7.0686 \: \text{ft}^2\).
This gives us the surface area of the sphere as approximately 7.0686 square feet. Understanding these steps helps break down the process, making it clear how we move from the geometric properties of the sphere to the numerical calculation of its surface area.