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Chapter 12: Problem 35

Give a geometric description of the following sets of points. $$x^{2}+y^{2}+z^{2}-8 x-14 y-18 z \leq 79$$

Short Answer

Expert verified
Answer: The geometric shape represented by the given inequality is a sphere with center (4, 7, 9) and a radius of 15. All points within the sphere satisfy the inequality.

Step by step solution

01

Rewrite inequality by completing the square

To rewrite the inequality by completing the square, we will rewrite it in the form of \((x-a)^2+(y-b)^2+(z-c)^2 \leq r^2\): $$x^{2}+y^{2}+z^{2}-8 x-14 y-18 z \leq 79$$ To complete the square, we need to find the square of the half of the coefficient of \(x\), \(y\), and \(z\) terms: For \(x\), \((-8/2)^2 = 16\); For \(y\), \((-14/2)^2 = 49\); For \(z\), \((-18/2)^2 = 81\). Now, add and subtract these values to the inequality: $$ (x^2-8x + 16) + (y^2-14y + 49) + (z^2 -18z + 81) \leq 79 + 16 + 49 + 81$$
02

Simplify inequality and rewrite as a standard equation

Now, we can group and simplify the inequality expression and rewrite it in the form of a standard equation: $$(x-4)^2+(y-7)^2+(z-9)^2 \leq 225$$
03

Interpret the geometric shape and its properties

Comparing this inequality to the standard form of the equation of a sphere, \((x-a)^2+(y-b)^2+(z-c)^2\leq r^2\), we can deduce that: The geometric shape represented by the inequality is a sphere with center \((a, b, c) = (4, 7, 9)\) and a radius \(r = 15\). All points within the sphere satisfy the inequality.

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Most popular questions from this chapter

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3,0\rangle, \mathbf{v}=\langle 1,1,1\rangle\)

Evaluate the following limits. $$\lim _{t \rightarrow 2}\left(\frac{t}{t^{2}+1} \mathbf{i}-4 e^{-t} \sin \pi t \mathbf{j}+\frac{1}{\sqrt{4 t+1}} \mathbf{k}\right)$$

Relationship between \(\mathbf{T}, \mathbf{N},\) and a Show that if an object accelerates in the sense that \(d^{2} s / d t^{2}>0\) and \(\kappa \neq 0,\) then the acceleration vector lies between \(\mathbf{T}\) and \(\mathbf{N}\) in the plane of \(\mathbf{T}\) and \(\mathbf{N}\). If an object decelerates in the sense that \(d^{2} s / d t^{2}<0,\) then the acceleration vector lies in the plane of \(\mathbf{T}\) and \(\mathbf{N},\) but not between \(\mathbf{T}\) and \(\mathbf{N}\)

Prove that for integers \(m\) and \(n\), the curve $$\mathbf{r}(t)=\langle a \sin m t \cos n t, b \sin m t \sin n t, c \cos m t\rangle$$ lies on the surface of a sphere provided \(a^{2}+b^{2}=c^{2}\).

The function \(f(x)=\sin n x,\) where \(n\) is a positive real number, has a local maximum at \(x=\pi /(2 n)\) Compute the curvature \(\kappa\) of \(f\) at this point. How does \(\kappa\) vary (if at all) as \(n\) varies?
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