Determine the Shape of Each Cross Section Formed by the Intersection Of

Home , Base (geometry), Cone, Cross section (geometry)

11-1 Cross Sections and Solids of Revolution

Determine the shape of each cross section 3. plane at an angle relative to the base through opposite formed by the intersection of the described faces plane with the solid. 1. plane perpendicular to the base

SOLUTION: A plane at an angle relative to the base through opposite faces will intersect 4 faces, any two SOLUTION: adjacent of which are perpendicular to each other. So the cross section is a rectangle. A plane perpendicular to the bases will intersect both bases on a straight line and the curved edge on two ANSWER: parallel lines. The resulting cross section is a rectangle rectangle. Describe the three-dimensional solid generated ANSWER: by rotating each two-dimensional shape around rectangle the given axis. 4. rectangle 2. plane parallel to the base

SOLUTION: Rotating a rectangle around a line that is along one of its sides will yield a cylinder.

ANSWER: SOLUTION: cylinder A plane parallel to the base of a triangular prism will intersect a cross section that is the same shape as its 5. circle bases. So the cross section is a triangle.

ANSWER: triangle SOLUTION: Rotating a circle around a line which it does not intersect will create a shape like a donut, which in math is called a torus.

ANSWER: torus (donut)

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6. triangle 8. plane at an angle relative to the base that intersects the base

SOLUTION: Rotating a right triangle around a line along one of its legs will create a right cone, where one leg is the radius and the other leg is the height of the cone. SOLUTION: A plane at an angle relative to the base that intersects ANSWER: the base of a rectangular pyramid will make a cross cone section that has four sides. The cross section will have one pair of parallel sides (the side along the Determine the shape of each cross section base of the pyramid and the opposite side along one formed by the intersection of the described triangular face) and one pair of sides that are not plane with the solid. parallel along two opposite triangular faces. So, the 7. plane at an angle relative to the bases that does not cross section is a trapezoid. intersect either base ANSWER: trapezoid

9. angled plane that intersects the sphere

SOLUTION: A plane at an angle relative to the bases that does not intersect either base will intersect only the lateral face of the cylinder. Since it is not parallel to either SOLUTION: base, and does not intersect either base, the shape of Any plane that intersects a sphere in more than one the cross section must be an ellipse. point will intersect the sphere with a cross section that is a circle. ANSWER: ellipse ANSWER: circle

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Describe the three-dimensional solid generated that the slices form each shape. by rotating each two-dimensional shape around a. rectangle the given axis. b. triangle c. trapezoid

10.

SOLUTION: Rotating a two dimensional arc around a line will create a hemisphere with a hemisphere cut out. This creates a bowl shape. SOLUTION: a. The cheese slice is in the shape of a triangular ANSWER: prism. The front, right or left view of the cheese slice bowl shape is a rectangle. So, to get a rectangular shape for the slice, one should cut it vertically. 11. rectangle

SOLUTION: A rectangle that is rotated around a line parallel to one of its sides that does not intersect the rectangle b. The top view of the cheese slice is a triangle. So, will create a cylinder with a cylinder cut out of its to get a triangular shape for the slice, one should cut center. This could be called an open cylinder, or a it horizontally. tube.

ANSWER: open cylinder (tube)

12. exponential function

c. The right or left view of the cheese slice is a rectangle. So, to get a trapezoidal shape for the slice, SOLUTION: one should cut it at an angle. The figure that is rotated around the line has two radii which create circles at each end, and between them there is a curve which is not a straight line, so this is not the frustum of a cone, but a shape with a lateral face that curves like a bell or the end of a horn like a trumpet or trombone.

ANSWER: horn shape

13. FOOD Describe how the cheese can be sliced so ANSWER: eSolutions Manual - Powered by Cognero Page 3 11-1 Cross Sections and Solids of Revolution

a. slice vertically 16. If a plane intersects with a cube at a vertex of the b. slice horizontally cube, what is the shape of the cross section? Explain c. slice at an angle your answer. SOLUTION: Describe each cross section. a triangle; Three faces of the cube meet to form the vertex, so the cross section is a two-dimensional figure with three sides.

ANSWER: a triangle; Three faces of the cube meet to form the 14. vertex, so the cross section is a two-dimensional figure with three sides. SOLUTION: 17. UFO Tanya has a model of a UFO. Sketch a two- dimensional figure that could be rotated around an axis to produce a three-dimensional solid similar to the model. See the model on page 799.

SOLUTION: A two-dimensional figure that could make the UFO A vertical plane will cut the sphere into two parts model by a rotation would look like half of the picture. with a cross section of a circle. Something like the figure below should produce the model. ANSWER: circle

15.

SOLUTION:

ANSWER: Sample answer:

A horizontal plane passing through the vertex will cut the cone into two parts with a cross section of a triangle.

ANSWER: triangle

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18. DESIGN Describe how you could create a tube with SOLUTION: a length of 10 inches, a diameter of 2 inches, and a a. To get a circle, you want to make a cut parallel to thickness of inch by rotating a 2-D figure around the bases of the cylinder. an axis. Make a sketch and label it.

SOLUTION: Take a rectangle 10 inches long and one-quarter inch wide and rotate it around a horizontal axis, with the outer edge of the rectangle at a distance of 1 inch from the axis.

ANSWER: b. To get a longer rectangle, you want to make a cut Take a rectangle 10 inches long and one-quarter inch along the length of the cylinder, through the center. wide and rotate it around a horizontal axis, with the outer edge of the rectangle at a distance of 1 inch from the axis.

c. To get an oval, you want to make a cut along an angle to the cylinder, but don't cut through either 19. POTTERY A potter creates three-dimensional base. objects by shaping the clay as it spins on a potter’s wheel. Describe the line or curve that could be rotated around a vertical axis to produce the vase shown on page 799.

SOLUTION: Sample answer: The curve would have the same shape as the edge of the vase. The curve would look like the letter S stretched vertically.

ANSWER: Sample answer: The curve would have the same shape as the edge of the vase. The curve would look like the letter S stretched vertically.

20. COOKIES Michelle is making cookies with a d. To get a shorter rectangle, you want to make a cylindrical roll of cookie dough. Describe how she cut along the length of the cylinder, but don't cut can cut the cookie dough to make each shape. through the center.

a. circle b. longest rectangle c. oval d. shorter rectangle eSolutions Manual - Powered by Cognero Page 5 11-1 Cross Sections and Solids of Revolution

atoms are arranged in regular geometrical patterns. Sketch a cross section from a horizontal slice of each crystal. Then describe the rotational symmetry about the vertical axis. a. tetragonal

ANSWER: a. Make a vertical cut. b. Make a horizontal through the center of the bases. c. Make a diagonal cut not through the bases. d. Make a horizontal cut not through the center of the b. hexagonal bases.

21. ART A piece of clay in the shape of a rectangular prism is cut in half as shown at the right. a. Describe the shape of the cross section. b. Describe how the clay could be cut to make the c. monoclinic cross section a triangle.

SOLUTION: SOLUTION: a. a. The cross section from a horizontal slice will look just like the top view of the figure, which appears to be a square.

The front view of the prism is a rectangle, so when it is cut vertically, the cross section will be a rectangle. Like all squares, a 90° rotation will produce an image b. identical to the preimage. The crystal appears the same for every 90° rotation about the axis.

b. The cross section from a horizontal slice will look just like the top view of the figure, which appears to be a regular hexagon. Three edges meet at each vertex. So, if you cut off a corner of the clay, you get a triangular cross section.

ANSWER: a. rectangle Like all regular hexagons, a 60° rotation will produce b. Cut off a corner of the clay. an image identical to the preimage. The crystal appears the same for every 60° rotation about the 22. EARTH SCIENCE Crystals are solids in which the eSolutions Manual - Powered by Cognero Page 6 11-1 Cross Sections and Solids of Revolution

axis. 24. CRITIQUE ARGUMENTS Ellen says that if you slice a sphere at an angle, you get an elliptical cross c. The cross section from a horizontal slice will look section. Is she correct? Explain your answer. just like the top view of the figure, which appears to be a rectangle with 2 triangular endpoints.

The crystal appears the same for every 180° rotation about the axis. SOLUTION: ANSWER: No, the cross section of a sphere is always a circle, a. not an ellipse.

ANSWER: No, the cross section of a sphere is always a circle, not an ellipse. The crystal appears the same for every 90° rotation about the axis. b.

The crystal appears the same for every 60° rotation about the axis. c.

The crystal appears the same for every 180° rotation about the axis.

23. Which of the following chess pieces can be created by rotating a two-dimensional figure around a vertical axis?

SOLUTION: Only the pawn, bishop, queen, and rook have rotational symmetry around a line.

ANSWER: pawn, bishop, queen, and rook eSolutions Manual - Powered by Cognero Page 7 11-1 Cross Sections and Solids of Revolution

25. REASONING If you slice off the top of a cone, you 26. CHALLENGE If you slice a cone parallel to the are left with a truncated cone. What two- base, the cross section is a circle. If the plane cuts at dimensional figure could be rotated around the axis to an angle through both sides of the cone, the cross produce a truncated cone? Name and sketch the section is an ellipse. What if the plane cuts at an figure. angle through the side of the cone and through the base of the cone? How are such cross sections different from a circle or an ellipse? Research conic sections and describe each cross section of a cone in terms of the features of each conic section.

SOLUTION: The top and bottom of the truncated cone are circles, which must be represented as straight lines perpendicular to the axis of rotation. Since the top circle is smaller than the bottom circle, one side of the figure should be a diagonal line that connects the top and bottom, the other side should be vertical along the axis. SOLUTION: The figure described is a right trapezoid like below. Sample answer: If the plane cuts at an angle through the base the cross section includes a straight line along the base, so the cross section cannot be a circle or ellipse. If the plane is at an angle, the curved edges will make part of a parabola. If the plane is perpendicular to the base, the curved edge will make ANSWER: part of a hyperbola. a right trapezoid ANSWER: Sample answer: If the plane cuts at an angle through the base the cross section includes a straight line along the base, so the cross section cannot be a circle or ellipse. If the plane is at an angle, the curved edges will make part of a parabola. If the plane is perpendicular to the base, the curved edge will make part of a hyperbola.

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Describe each cross section.

29. 27. SOLUTION: SOLUTION: The base of the cone is a circle. So, a plane parallel A vertical plane will cut the prism into two parts with to the base will give a cross section of a circle. a cross section of a rectangle.

ANSWER: ANSWER: circle rectangle

28. 30.

SOLUTION: SOLUTION: The bases of the prism are hexagons. So, a plane A vertical plane will cut the pyramid into two parts parallel to the base will give a cross section of a with a cross section of a triangle. hexagon.

ANSWER: ANSWER: hexagon triangle

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31. Which of the following three-dimensional solids can 33. Eduardo has a piece of wood in the shape of a square be generated by rotating a two-dimensional figure pyramid. The sides of the base are 6 inches long. around an axis? Eduardo cuts the pyramid with a single straight cut parallel to the base. Which of the following A pyramid statements about the cross section must be true? B banana C cube A The cross section is a square with an area less D egg than 36 in². B The cross section is a square with an area of 36 SOLUTION: in². A pyramid and a cube do not have rounded edges so C The cross section is a triangle with an area less they can be eliminated. than 36 in². A banana does not have rotational symmetry around D The cross section is a triangle with an area of 36 an axis. in². An egg does have rotational symmetry around an axis. SOLUTION: The correct choice is D. Visualize a square pyramid with a plane passing through it which is parallel to the base of the pyramid. ANSWER: The cross section formed would be the green square D shown in the figure below. 32. Which of the following shapes could not be a cross section of the prism?

A octagon The area of the base of the pyramid is or 36 in². B pentagon Any square formed by the cross section of this C rectangle pyramid, that isn't the base, will have a smaller area D triangle than the base since the lateral faces of the pyramid SOLUTION: slope in until they meet at the Vertex. Therefore, the An octagon cannot be the cross section of a correct choice is A. rectangular prism, because a rectangular prism has ANSWER: only 6 faces and an octagon has 8 sides. A

The correct choice is A. 34. What shape is generated by rotating a rectangle around an axis parallel to one of its sides? ANSWER: A SOLUTION: Rotating a rectangle around a line parallel to one of its sides will create a cylinder or a cylindrical tube.

ANSWER: cylinder eSolutions Manual - Powered by Cognero Page 10 11-1 Cross Sections and Solids of Revolution

35. Describe how to generate a torus, or donut shape, with an outside radius of 6 inches.

SOLUTION: To make a torus or a donut shape consider what its cross section looks like, which is two circles.

Rotate a circle around an axis. The outer edge of the circle should be 6 inches from the axis. a. The pyramid is intersected by a plane perpendicular to the base of the pyramid and through ANSWER: the vertex. Determine the shape of the cross section. Rotate a circle around an axis. The outer edge of the circle should be 6 inches from the axis. b. What is the area of the cross section? 36. Describe how to generate a tube with the following dimensions: c. A plane parallel to the base of the pyramid intersects the pyramid at a height of 12 centimeters. length = 20 feet Determine the shape of the cross section. diameter = 8 inches thickness = 0.5 inches d. What is the area of the cross section?

SOLUTION: e. A third plane intersects the pyramid at an angle. It To make a tube that is 20 feet long with a diameter of goes through one face of the pyramid at a height of 8 inches and a thickness of 0.5 inches, start with a 12 centimeters, and it intersects theedge of the rectangle with ℓ = 20 feet and w = 0.5 inch. Rotate opposite side of the base. Determine the shape of the the rectangle about an axis 4 inches from the outer cross section. edge of the rectangle. SOLUTION: ANSWER: a. The shape of the cross section with a plane Start with a rectangle with ℓ = 20 feet and w = 0.5 intersecting the pyramid perpendicular to the base inch. Rotate the rectangle about an axis 4 inches and through the vertex will make a triangle with the from the outer edge of the rectangle. slant height as two sides and the other side is part of the base. 37. Describe two ways to intersect a solid with a plane to b. The area of that triangle is (0.5)(24 cm)(24 cm) = produce a cross section that is a trapezoid. 288 cm2 SOLUTION: c. The shape of a cross section parallel to the base 12 Sample answer: A plane intersects a square pyramid centimeters above the base will be a square. at an angle through opposite faces. d. That square will have sides that are 12 centimeters A plane intersects a triangular pyramid at an angle long, because the base and height of the pyramid are through one lateral face and through the base. congruent, so the area is: (12 cm)(12 cm) = 144 cm2 ANSWER: e. The shape of the cross section with a plane at an Sample answer: A plane intersects a square pyramid angle that intersects the middle of one lateral face at an angle through opposite faces. and the base will be a trapezoid, because it has two A plane intersects a triangular pyramid at an angle parallel faces that are not of equal length. through one lateral face and through the base.

38. MULTI-STEP The figure below is a square pyramid ANSWER: with a height of 24 centimeters and a base length of a. triangle 24 centimeters. eSolutions Manual - Powered by Cognero Page 11 11-1 Cross Sections and Solids of Revolution

b. 288 cm2 c. square d. 144 cm2 e. trapezoid