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Chapter 11: Problem 24

Find an equation or inequality that describes the following objects. A sphere with center (1,2,0) passing through the point (3,4,5)

Short Answer

Expert verified
The equation of the sphere is: (x-1)^2 + (y-2)^2 + z^2 = 29. The inequality describing the region inside the sphere is: (x-1)^2 + (y-2)^2 + z^2 鈮� 29.

Step by step solution

01

Find the Radius

By applying the distance formula between the center (h,k,l) = (1,2,0) and the given point (x,y,z) = (3,4,5), we will find the radius, r: $$ r = \sqrt{(x-h)^2 + (y-k)^2 + (z-l)^2} = \sqrt{(3-1)^2 + (4-2)^2 + (5-0)^2} $$
02

Evaluate the Radius

Now, compute the numerical value of the radius: $$ r = \sqrt{(2)^2 + (2)^2 + (5)^2} = \sqrt{29} $$ The radius, r, is equal to \(\sqrt{29}\).
03

Find the Equation of the Sphere

Now, we can write the general equation of a sphere centered at (h,k,l) with radius r: $$ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 $$ Now, replace h, k, l, and r with the values we found earlier: $$ (x-1)^2 + (y-2)^2 + (z-0)^2 = (\sqrt{29})^2 $$
04

Simplify the Equation

Finally, simplify the given equation: $$ (x-1)^2 + (y-2)^2 + z^2 = 29 $$ This is the equation of the sphere. To convert it into an inequality, just change the equals sign to either '>=' for outside the sphere or '<=' for inside the sphere: $$ (x-1)^2 + (y-2)^2 + z^2 \leq 29 $$ This inequality describes the region inside the sphere with center (1,2,0) and passing through the point (3,4,5).

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Key Concepts

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

Distance Formula
The Distance Formula is a tool used to calculate the distance between two points in space. Imagine needing to measure how far one point is from another. Given two points, \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\), the formula to find the distance between them is:
  • \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\)

This formula helps us compute the radius of the sphere when we know the center and a point on its surface.
Radius Calculation
Calculating the radius of a sphere involves determining how far a point on its surface is from its center. With the center at (1,2,0) and the point (3,4,5) on the sphere, we use the Distance Formula:
  • \(r = \sqrt{(3-1)^2 + (4-2)^2 + (5-0)^2}\)
  • This simplifies to \(r = \sqrt{4 + 4 + 25} = \sqrt{29}\)

Thus, the radius of the sphere is \(\sqrt{29}\), which is essential for defining the sphere's equation.
Equation of a Sphere
The Equation of a Sphere represents a set of all points in space that are at the same distance (radius) from the center. The general form of this equation, where \(h, k, l\) is the center and \(r\) is the radius, is:
  • \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\)

For our sphere, with center (1,2,0) and radius \(\sqrt{29}\), the equation becomes:
  • \((x-1)^2 + (y-2)^2 + z^2 = 29\)
Inequality Representation
An Inequality Representation of a sphere describes a region in space, rather than just the surface. If you imagine a sphere, the inequality can specify either the inside or the outside area:
  • \((x-1)^2 + (y-2)^2 + z^2 \leq 29\) represents points inside or on the sphere.
  • \((x-1)^2 + (y-2)^2 + z^2 \geq 29\) would describe points on or outside the sphere.

This form of representation is useful in problems involving volume or constraints within a sphere.

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Most popular questions from this chapter

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and \(f\) are real numbers. It can be shown that this curve lies in a plane. Find a general expression for a nonzero vector orthogonal to the plane containing the curve. $$\begin{aligned}\mathbf{r}(t)=&(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j} \\ &+(e \cos t+f \sin t) \mathbf{k},\end{aligned}$$ where \(\langle a, c, e\rangle \times\langle b, d, f\rangle \neq \mathbf{0}\).

Suppose the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is smooth on an interval containing the point \(t_{0} .\) The line tangent to \(\mathbf{r}(t)\) at \(t=t_{0}\) is the line parallel to the tangent vector \(\mathbf{r}^{\prime}\left(t_{0}\right)\) that passes through \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right) .\) For each of the following functions, find an equation of the line tangent to the curve at \(t=t_{0} .\) Choose an orientation for the line that is the same as the direction of \(\mathbf{r}^{\prime}\). $$\mathbf{r}(t)=\langle 2+\cos t, 3+\sin 2 t, t\rangle ; t_{0}=\pi / 2$$

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=t e^{t} \mathbf{i}+t \sin t^{2} \mathbf{j}-\frac{2 t}{\sqrt{t^{2}+4}} \mathbf{k}$$

Jack pulls east on a rope attached to a camel with a force of 40 ib. Jill pulls north on a rope attached to the same camel with a force of 30 Ib. What is the magnitude and direction of the force on the camel? Assume the vectors lie in a horizontal plane.

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$a(\mathbf{u}+\mathbf{v})=a \mathbf{u}+a \mathbf{v}$$
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