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Chapter 10: Problem 27

Identify the geometric shape described by the given equation. $$(x-1)^{2}+y^{2}+(z+2)^{2}=4$$

Short Answer

Expert verified
The geometric shape described by the equation \((x-1)^{2}+y^{2}+(z+2)^{2}=4\) is a sphere with center at \((1, 0, -2)\) and radius 2.

Step by step solution

01

Identifying the Form of the Equation

The equation \((x-1)^{2}+y^{2}+(z+2)^{2}=4\) is in the form of \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), which is the standard form of the equation of a sphere in 3-dimensional space.
02

Finding the Center of the Sphere

The center \((h, k, l)\) of the sphere is given by the values that make each part of the equation zero. Comparing to the standard form, we can see that \(h=1\), \(k=0\), and \(l=-2\). Hence, the center of the sphere is at \((1, 0, -2)\).
03

Finding the Radius of the Sphere

The radius \(r\) of the sphere can directly be obtained from the equation as the square root of the constant term on the right. So, \(r = \sqrt{4} = 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3-dimensional geometry
3-dimensional geometry involves shapes and figures that exist in three dimensions: length, width, and height. Unlike 2-dimensional shapes that lie flat on a plane, 3D shapes have depth, which allows them to occupy space. This dimension introduces volume and makes geometry richer and more complex.
A sphere is a classic example of a 3-dimensional object. It is perfectly symmetrical, with all points on its surface equidistant from its center. Understanding these 3D shapes requires looking at equations like the standard form for spheres, which helps us locate and describe the object in space.
  • The equation \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\) represents a sphere in 3D geometry.
  • Here, \(h, k, l\) are the coordinates of the sphere's center.
  • \(r\) is the radius, indicating the distance from the center to any point on the sphere's surface.
Recognizing these forms is essential in solving 3D geometry problems effectively.
Center of a sphere
The center of a sphere in mathematics is a critical point that helps define the shape and position of the sphere in 3D space. In the equation \(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), the center is represented by the coordinates \(h, k, l\). This point is crucial because:
  • It acts as the fixed equidistant point from which the sphere's surface extends.
  • Changing the center's coordinates shifts the entire sphere in space.
In our given exercise, the equation \(x-1)^{2}+y^{2}+(z+2)^{2}=4\) has been simplified to find that the center of the sphere is at \(1, 0, -2\).
This tells us that from the origin of the coordinate system, the center is 1 unit along the x-axis, at the origin along the y-axis, and 2 units backward along the z-axis. Understanding how to find the center allows us to graph and manipulate spheres effectively.
Radius of a sphere
The radius of a sphere is the distance from its center to any point on the sphere's surface. This distance is consistent throughout the surface of the sphere, reflecting its uniformity. In the sphere's equation \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), the radius \(r\) is derived from the right side of the equation as the square root of the constant.
In the exercise equation \(x-1)^{2}+y^{2}+(z+2)^{2}=4\), the radius is straightforward to calculate as \(\) the square root of 4. This gives \(r = 2\).
  • The radius is vital for defining the size of the sphere.
  • It helps in calculating the volume and surface area.
Getting familiar with finding the radius aids in various applications, from basic graphing to complex engineering and physics calculations.

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Most popular questions from this chapter

Use the Cauchy-Schwartz Inequality in \(n\) dimensions to show that \(\left(\sum_{k=1}^{n}\left|a_{k} b_{k}\right|\right)^{2} \leq\left(\sum_{k=1}^{n} a_{k}^{2}\right)\left(\sum_{k=1}^{n} b_{k}^{2}\right) .\) If both \(\sum_{k=1}^{\infty} a_{k}^{2}\) and \(\sum_{k=1}^{\infty} b_{k}^{2}\) converge what can be concluded? Apply the result to \(a_{k}=\frac{1}{k}\) and \(b_{k}=\frac{1}{k^{2}}\)

Among all sets of nonnegative numbers \(p_{1}, p_{2}, \ldots, p_{n}\) that sum to \(1,\) find the choice of \(p_{1}, p_{2}, \ldots, p_{n}\) that minimizes \(\sum_{k=1}^{n} p_{k}^{2}\)

Sketch the given plane. $$x=4$$

Sketch the appropriate traces, and then sketch and identify the surface. $$9 x^{2}+z^{2}=9$$

Find the intersection of the planes. $$3 x+y-z=2 \text { and } 2 x-3 y+z=-1$$
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