In each part, write an equation for the surface.(a) The plane that contains the x-axis

The distance between the points \((1,-2,0)\) and \((4,0,5)\) is _______. Equation Transcription: Text Transcription: (1, -2, 0) (4, 0, 5)

The graph of \((x-3)^{2}+(y-2)^{2}+(z+1)^{2}=16\) is a _______ of radius _______ centered at _______. Equation Transcription: Text Transcription: (x-3)^2 + (y-2)^2 + (z+1)^2 = 16

The shortest distance from the point \((4,0,5)\) to the sphere \((x-1)^{2}+(y+2)^{2}+z^{2}=36\) is _______. Equation Transcription: Text Transcription: (4, 0, 5) (x-1)^2 + (y+2)^2 + z^2 = 36

Let \(S\) be the graph of \(x^{2}+z^{2}+6 z=16\) in 3-space. (a) The intersection of S with the xz-plane is a circle with center _______ and radius _______. (b) The intersection of S with the xy-plane is two lines, x = _______ and x = _______. (c) The intersection of S with the yz-plane is two lines, z = _______ and z = _______. Equation Transcription: Text Transcription: S x^2 + z^2 +6z = 16

Interpret the graph of \(x=1\) in the contexts of (a) a number line (b) 2-space (c) 3-space. Equation Transcription: Text Transcription: x = 1

Consider the points \(\mathrm{P}(3,1,0) \text { and } \mathrm{Q}(1,4,4)\). (a) Sketch the triangle with vertices \(\mathrm{P}, \mathrm{Q}, \text { and }(1,4,0)\). Without computing distances, explain why this triangle is a right triangle, and then apply the Theorem of Pythagoras twice to find the distance from P to Q. (b) Repeat part (a) using the points \(P, Q, \text { and }(3,4,0)\). (c) Repeat part (a) using the points \(P, Q, \text { and }(1,1,4)\). Equation Transcription: Text Transcription: P (3, 1, 0) Q (1, 4, 4) P, Q, and (1, 4, 0) P, Q, and (3, 4, 0) P, Q, and (1, 1, 4)

(a) Consider a box whose sides have lengths a, b, and c. Use the Theorem of Pythagoras to show that a diagonal of the box has length \(d=\sqrt{a^{2}+b^{2}+c^{2}}\). [Hint: Use the Theorem of Pythagoras to find the length of a diagonal of the base and then again to find the length of a diagonal of the entire box.] (b) Use the result of part (a) to derive formula (2). Equation Transcription: Text Transcription: d = sqrt a^2 + b^2 + c^2

(a) Make a conjecture about the set of points in 3-space that are equidistant from the origin and the point \((1,0,0)\). (b) Confirm your conjecture in part (a) by using distance formula (2). Equation Transcription: Text Transcription: (1, 0, 0)

Find the center and radius of the sphere that has \((1,-2,4) \text { and }(3,4,-12)\) as endpoints of a diameter. [See Exercise 4.] Equation Transcription: Text Transcription: (1, -2, 4) (3, 4, -12)

Show that \((4,5,2),(1,7,3), \text { and }(2,4,5)\) are vertices of an equilateral triangle. Equation Transcription: Text Transcription: (4, 5, 2) (1, 7, 3) (2, 4, 5)

(a) Show that \((2,1,6),(4,7,9), \text { and }(8,5,-6)\) are the vertices of a right triangle. (b) Which vertex is at the \(90^{\circ}\) angle? (c) Find the area of the triangle. Equation Transcription: Text Transcription: (2, 1, 6), (4, 7, 9), (8, 5, -6) 90 degrees

Find the distance from the point \((-5,2,-3)\) to the (a) xy-plane (b) xz-plane (c) yz-plane (d) x-axis (e) y-axis (f ) z-axis. Equation Transcription: Text Transcription: (-5, 2, -3)

In each part, find the standard equation of the sphere that satisfies the stated conditions. (a) Center \((7,1,1) ; \text { radius }=4 \text {. }\). (b) Center \((1,0,-1) ; \text { diameter }=8\). (c) Center \((-1,3,2)\) and passing through the origin. (d) A diameter has endpoints \((-1,2,1) \text { and }(0,2,3)\). Equation Transcription: Text Transcription: (7, 1, 1); radius = 4 (1, 0, -1); diameter = 8 (-1, 3, 2) (-1, 2, 1) and (0, 2, 3)

Find equations of two spheres that are centered at the origin and are tangent to the sphere of radius 1 centered at \((3,-2,4)\). Equation Transcription: Text Transcription: (3, -2, 4)

In each part, find an equation of the sphere with center \((2,-1,-3)\) and satisfying the given condition. (a) \(\text { Tangent to the } \mathrm{xy} \text {-plane }\) (b) \(\text { Tangent to the xz-plane }\) (c) \(\text { Tangent to the yz-plane }\) Equation Transcription: Text Transcription: (2, -1, -3) Tangent to the xy-plane Tangent to the xz-plane Tangent to the yz-plane

(a) Find an equation of the sphere that is inscribed in the cube that is centered at the point \((-2,1,3)\) and has sides of length 1 that are parallel to the coordinate planes. (b) Find an equation of the sphere that is circumscribed about the cube in part (a). (c) Find an equation of the sphere that is inscribed in the cube determined by the planes \(x=6, x=2, y=5, y=9, z=0, \text { and } z=4\). (d) Find an equation of the sphere that is circumscribed about the cube in part (c). Equation Transcription: Text Transcription: (-2, 1, 3) x = 6, x = 2, y = 5, y = 9, z = 0, and z = 4

A sphere has center in the first octant and is tangent to each of the three coordinate planes. Show that the center of the sphere is at a point of the form \((r, r, r)\), where r is the radius of the sphere. Equation Transcription: Text Transcription: (r, r, r)

A sphere has center in the first octant and is tangent to each of the three coordinate planes. The distance from the origin to the sphere is \(3-\sqrt{3}\) units. Find an equation for the sphere. Equation Transcription: Text Transcription: 3 - sqrt 3

1922 TrueFalse Determine whether the statement is true or false. Explain your answer. By definition, a cylindrical surface is a right circular cylinder whose axis is parallel to one of the coordinate axes.

Determine whether the statement is true or false. Explain your answer. The graph of \(x^{2}+y^{2}=1\) in 3-space is a circle of radius 1 centered at the origin. Equation Transcription: Text Transcription: x^2 + y^2 = 1

Determine whether the statement is true or false. Explain your answer. If a point belongs to both the \(\mathrm{xy} \text {-plane }\) and the \(\text { xz-plane }\), then the point lies on the \(\text { X-axis }\). Equation Transcription: Text Transcription: xy - plane xz - plane x - axis

Determine whether the statement is true or false. Explain your answer. A sphere with center \(P\left(x_{0}, y_{0}, z_{0}\right)\) and radius \(r\) consists of all points \((\mathrm{x}, \mathrm{y}, \mathrm{z})\) that satisfy the inequality \(\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2} \leq r^{2}\) Equation Transcription: Text Transcription: P(x_0, y_0, z_0) r (x, y, z) (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 leq r^2

Describe the surface whose equation is given. \(x^{2}+y^{2}+z^{2}+10 x+4 y+2 z-19=0\) Equation Transcription: Text Transcription: x^2 + y^2 +z^2 +10x +4y + 2z - 19 = 0

Describe the surface whose equation is given. \(x^{2}+y^{2}+z^{2}-y=0\) Equation Transcription: Text Transcription: x^2 + y^2 + z^2 - y = 0

Describe the surface whose equation is given. \(2 x^{2}+2 y^{2}+2 z^{2}-2 x-3 y+5 z-2=0\) Equation Transcription: Text Transcription: 2x^2 + 2y^2 +2z^2 - 2x -3y +5z - 2 = 0

Describe the surface whose equation is given. \(x^{2}+y^{2}+z^{2}+2 x-2 y+2 z+3=0\) Equation Transcription: Text Transcription: x^2 + y^2 +z^2 +2x - 2y + 2z + 3 = 0

Describe the surface whose equation is given. \(x^{2}+y^{2}+z^{2}-3 x+4 y-8 z+25=0\) Equation Transcription: Text Transcription: x^2 + y^2 + z^2 ? 3x + 4y ? 8z + 25 = 0

Describe the surface whose equation is given. \(x^{2}+y^{2}+z^{2}-2 x-6 y-8 z+1=0\) Equation Transcription: Text Transcription: x^2 + y^2 + z^2 ? 2x ? 6y ? 8z + 1 = 0

In each part, sketch the portion of the surface that lies in the first octant. (a) \(y=x\) (b) \(y = z\) (c) \(x = z\) Equation Transcription: Text Transcription: y = x y = z x = z

In each part, sketch the graph of the equation in 3-space. (a) \(x=1\) (b) \(y = 1\) (c) \(z = 1\) Equation Transcription: Text Transcription: x = 1 y = 1 z = 1

In each part, sketch the graph of the equation in 3-space. (a) \(x^{2}+y^{2}=25\) (b) \(y^{2}+z^{2}=25\) (c) \(x^{2}+z^{2}=25\) Equation Transcription: Text Transcription: x^2 + y^2 = 25 y^2 + z^2 = 25 x^2 + z^2 = 25

In each part, sketch the graph of the equation in 3-space. (a) \(x=y^{2}\) (b) \(z=x^{2}\) (c) \(y=z^{2}\) Equation Transcription: Text Transcription: x = y^2 z = x^2 y = z^2

In each part, write an equation for the surface. (a) The plane that contains the \(x \text {-axis and the point }(0,1,2)\). (b) The plane that contains the \(y \text {-axis and the point }(1,0,2)\). (c) The right circular cylinder that has \(\text { radius } 1\) and is centered on the line parallel to the \(\text { z-axis }\) that passes through the point \((1, 1, 0)\). (d) The right circular cylinder that has \(\text { radius } 1\) and is centered on the line parallel to the \(\text { y-axis }\) that passes through the point \((1, 0, 1)\). Equation Transcription: Text Transcription: x-axis and the point (0, 1, 2). y-axis and the point (1, 0, 2) radius 1 z-axis (1, 1, 0) y-axis (1, 0, 1)

Find equations for the following right circular cylinders. Each cylinder has \(radius a\) and is tangent to two coordinate planes. Equation Transcription: Text Transcription: radius a

Sketch the surface in 3-space. \(y=\sin x\) Equation Transcription: Text Transcription: y = sin x

Sketch the surface in 3-space. \(y=e^{x}\) Equation Transcription: Text Transcription: y = e^x

Sketch the surface in 3-space. \(z=1-y^{2}\) Equation Transcription: Text Transcription: z = 1 - y^2

Sketch the surface in \(3\)-space. \(z=\cos x\) Equation Transcription: 3 z = cos x Text Transcription: 3 z = cos x

Sketch the surface in \(3\)-space. \(2 x+z=3\) Equation Transcription: 3 2x + z = 3 Text Transcription: 3 2x + z = 3

Sketch the surface in \(3\)-space. \(2 x+3 y=6\) Equation Transcription: 3 2x + 3y = 6 Text Transcription: 3 2x + 3y = 6

Sketch the surface in \(3\)-space. \(4 x^{2}+9 z^{2}=36\) Equation Transcription: 3 Text Transcription: 3 4x^2 + 9z^2 = 36

Sketch the surface in \(3\)-space. \(z=\sqrt{3-x}\) Equation Transcription: 3 z = Text Transcription: 3 z = sqrt 3 - x

Sketch the surface in \(3\)-space. \(y^{2}-4 z^{2}=4\) Equation Transcription: 3 Text Transcription: 3 y^2 ? 4z^2 = 4

Sketch the surface in \(3\)-space. \(\mathrm{yz}=1\) Equation Transcription: 3 yz = 1 Text Transcription: 3 yz = 1

Use a graphing utility to generate the curve \(y=x^{3} /\left(1+x^{2}\right)\) in the xy-plane, and then use the graph to help sketch the surface \(z=y^{3} /\left(1+y^{2}\right)\) in \(3\)-space. Equation Transcription: 3 Text Transcription: y = x^3/(1 + x^2) z = y^3/(1 + y^2) 3

Use a graphing utility to generate the curve \(y=x /\left(1+x^{4}\right)\) in the xy-plane, and then use the graph to help sketch the surface \(z=y /\left(1+y^{4}\right)\) in \(3\)-space. Equation Transcription: 3 Text Transcription: y = x/(1 + x^4) z = y/(1 + y^4) 3

If a bug walks on the sphere \(x^{2}+y^{2}+z^{2}+2 x-2 y-4 z-3=0\) Equation Transcription: Text Transcription: x^2 + y^2 + z^2 + 2x - 2y - 4z - 3 = 0

Describe the set of all points in \(3\)-space whose coordinates satisfy the inequality \(x^{2}+y^{2}+z^{2}-2 x+8 z \leq 8\). Equation Transcription: 3 Text Transcription: 3 x^2 + y^2 + z^2 ? 2x + 8z leq 8

Describe the set of all points in \(3\)-space whose coordinates satisfy the inequality \(y^{2}+z^{2}+6 y-4 z>3\). Equation Transcription: 3 Text Transcription: 3 y^2 + z^2 + 6y ? 4z > 3

The distance between a point \(P(x, y, z)\) and the point \(A(1,?2, 0)\) is twice the distance between \(P\) and the point \(B(0, 1, 1)\). Show that the set of all such points is a sphere, and find the center and radius of the sphere. Equation Transcription: P(x, y, z) A(1,?2, 0) P B(0, 1, 1) Text Transcription: P(x, y, z) A(1,?2, 0) P B(0, 1, 1)

As shown in the accompanying figure, a bowling ball of radius \(R\) is placed inside a box just large enough to hold it, and it is secured for shipping by packing a Styrofoam sphere into each corner of the box. Find the radius of the largest Styrofoam sphere that can be used. [Hint: Take the origin of a Cartesian coordinate system at a corner of the box with the coordinate axes along the edges.] Equation Transcription: R Text Transcription: R

Consider the equation \(x^{2}+y^{2}+z^{2}+G x+H y+l z+J=0\) and let \(\mathrm{K}=\mathrm{G}^{2}+\mathrm{H}^{2}+\mathrm{I}^{2}-4 \mathrm{J}\). (a) Prove that the equation represents a sphere if \(\mathrm{K}>0\), a point if \(K=0\), and has no graph if\(\mathrm{K}<0\). (b) In the case where \(\mathrm{K}>0\), find the center and radius of the sphere. Equation Transcription: K > 0 K = 0 K < 0 Text Transcription: x^2 + y^2 + z^2 + Gx + Hy + Iz + J = 0 K = G^2 + H^2 + I^2 ? 4J K > 0 K = 0 K < 0

(a) The accompanying figure shows a surface of revolution that is generated by revolving the curve \(y=f(x)\) in the xy-plane about the x-axis. Show that the equation of this surface is \(y^{2}+z^{2}=[f(x)]^{2}\). [Hint: Each point on the curve traces a circle as it revolves about the x-axis.] (b) Find an equation of the surface of revolution that is generated by revolving the curve \(y=e^{x}\) in the xy-plane about the x-axis. (c) Show that the ellipsoid \(3 x^{2}+4 y^{2}+4 z^{2}=16\) is a surface of revolution about the x-axis by finding a curve \(y=f(x)\) in the xy-plane that generates it. Equation Transcription: y = f(x) Text Transcription: y = f(x) y^2 + z^2 = [f(x)]^2 y = e^x 3x^2 + 4y^2 + 4z^2 = 16

In each part, use the idea in Exercise \(53(a)\) to derive a formula for the stated surface of revolution. (a) The surface generated by revolving the curve \(x=f(y)\) in the xy-plane about the y-axis. (b) The surface generated by revolving the curve \(y=f(z)\) in the yz-plane about the z-axis. (c) The surface generated by revolving the curve \(z=f(x)\) in the xz-plane about the x-axis. Equation Transcription: 53(a) x = f(y) y = f(z) z = f(x) Text Transcription: 53(a) x = f(y) y = f(z) z = f(x)

Show that for all values of \(\theta\) and \(\phi\), the point \((a \sin \phi \cos \theta, a \sin \phi \sin \theta, a \cos \phi)\) lies on the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\). Equation Transcription: ???? (a sin ???? cos , a sin ???? sin , a cos ????) Text Transcription: theta phi (a sin phi cos theta, a sin phi sin theta, a cos phi) x^2 + y^2 + z^2 = a^2

Writing Explain how you might determine whether a set of points in \(3\)-space is the graph of an equation involving at most two of the variables \(x\), \(y\), and \(z\). Equation Transcription: 3 x y z Text Transcription: 3 x y z

Writing Discuss what happens geometrically when equations in \(x\), \(y\), and \(z\) are replaced by inequalities. For example, compare the graph of \(x^{2}+y^{2}+z^{2}=1\) with the set of points that satisfy the inequality \(x^{2}+y^{2}+z^{2} \leq 1\). Equation Transcription: x y z Text Transcription: x y z x^2 + y^2 + z^2 = 1 x^2 + y^2 + z^2 leq 1