Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Find the standard equation of the sphere. Center: \((4,-1,1) ;\) radius: 5

Step-by-Step Solution

Answer

The standard equation of the sphere is \((x - 4)^2 + (y + 1)^2 + (z - 1)^2 = 25\).

Step 1: Identify the coordinates of the center

Center coordinates are given as \((4, -1, 1)\). So, \(h = 4\), \(k = -1\), and \(l = 1\) in the sphere equation above.

Step 2: Identify the radius

The radius of the sphere is given as 5. So, \(r = 5\) in the sphere equation.

Step 3: Substitute the values into the formula

Substitute \(h = 4\), \(k = -1\), \(l = 1\), and \(r = 5\) into the standard sphere equation \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). This would result in the equation \((x - 4)^2 + (y + 1)^2 + (z - 1)^2 = 25\).

Key Concepts

3D Geometry

When we talk about 3D geometry, we are exploring objects that exist in three-dimensional space. This means they have length, width, and height, making them more complex than flat, two-dimensional shapes.
Let's take a moment to visualize working with 3D objects. Consider common shapes like cubes, pyramids, and spheres. A sphere, much like a 3D version of a circle, is perfectly round in every direction.

• Dimensions: Length, width, and height.
• Shape Representation: 3D shapes are like their 2D counterparts but with added depth.
• Real-World Examples: Balls, planets, and bubbles are everyday representations of spheres.

Working with 3D geometry, particularly spheres, can initially seem daunting, but it really is just about understanding how these dimensions interact to form space-filling objects. It's essential to get comfortable with coordinates along the x, y, and z axes, as these will allow you to precisely locate objects in three-dimensional space.

Equation of a Sphere

The equation of a sphere is a crucial part of understanding 3D shapes. It allows us to express a sphere's size and position in space in a concise mathematical form. Here's the standard equation:\[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]This equation might look complex at first glance, but it's actually quite straightforward.

• (x, y, z): Represents any point on the surface of the sphere.
• (h, k, l): These are the coordinates of the sphere's center.
• r: Stands for the radius of the sphere, explaining the sphere's size.

When you substitute particular numbers for h, k, l, and r, you get a specific sphere. This equation helps us understand how spheres change position and size by altering these values. Once you become familiar with this equation, you can apply it to many mathematical and physical problems, from simple geometry homework to complex 3D modeling.

Center and Radius of Sphere

To fully understand the structure of any sphere, you need to identify its center and radius. These two crucial components give the sphere its shape and position.The center of a sphere is its exact middle point in space. At this point, the distance is equal to the radius in all directions.

• Center (h, k, l): This is where you would "pin" the sphere if you were physically holding it in a model. For example, in our exercise, the center is at coordinates \(4, -1, 1\).
• Radius r: This measures how far the sphere stretches out from its center in every direction. In our exercise, the radius is 5.

These constants not only define the sphere but also affect how it interacts with its environment. If you imagine moving the center or changing the radius in real life, you see shifts in the position and size of the sphere. This makes understanding centers and radii crucial when visualizing and solving problems in 3D geometry.