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) eSolutions Manual - Powered by Cognero Page 1 11-1 Cross Sections and Solids of Revolution 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 eSolutions Manual - Powered by Cognero Page 2 11-1 Cross Sections and Solids of Revolution 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 eSolutions Manual - Powered by Cognero Page 4 11-1 Cross Sections and Solids of Revolution 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.