Chapter 1 Test Review Question Answers 1. Find the total surface area and volume of a cube in which the diagonal measures yards. x + x ) = ) x = x x A T.S. = bh) = ) ) = 1 yd V = BH = bh)h = ) ) ) = yd. A right cylindrical hole of diameter inches is drilled out of a cube with an edge of 7 inches. Find the surface area and volume of the remaining figure. A T.S. = A Prism - A Circle ) + A L.S. of the Inside Cylinder = bh)h - πr + πr)h ) = 7) 7) - π + π) 7) = - π + π 7 7 = + π ) in 7 V = V Prism - V Cylinder = bh)h - πr )H ) = 7) 7) 7) - π) 7) = - 17π ) in Baroody Page 1 of 11
Chapter 1 Test Review Question Answers. Find the slant height and surface area of a right circular cone with a volume of π cubic inches and a base circumference of 10π inches. C = 10π c = + 7 r = c = 7 c 7 V = 1 πr )H A T.S. = πr + πrl C Base = 10π π = 1 π)h 7 = H = π + ) 7)π = π + π 7) in. Find the volume and total surface area of a regular square pyramid that has an edge height of 1 yards and a side of the base of 10 yards. H = 1 - = 11 1 1 H A T.S. = 10 + 1 ) bh = 100 + 10) 1) = 0 yd V = 1 BH = 1 10) 10) 11 ) = 100 11 ) yd 10. The volume of a sphere in cubic feet is numerically equal to four times its surface area in square feet. What is the radius of the sphere? V Sphere = A T.S of Sphere )!r ) =!r ) r = 1r r = 1 ft Baroody Page of 11
Chapter 1 Test Review Question Answers. 7. Find the total surface area and volume of a regular tetrahedron if every edge measures feet. H H = ) ) - Base of Pyramid = - 1 = 1 ) A T.S. = s = ) u V = 1 s )H = ) ) = 1 1 ) = 1 1 1 = ) ft A frustum of a right cone has a slant height of feet. The radius of the bottom base is 1 feet and the radius of the top base is feet. Find the volume and total surface area of the frustum. Now, the small to the large by AA, so 1 = x x + x = 1 1 1 V Frustum = V Large Cone - V Smaller Cone = 1!R )H Large Cone - 1!r )H Small Cone = 1 1! ) 1 ) - 1 1! ) ) = 7! -! =! ft A T.S. =!RL -!rl +!R +!r =! 1) ) -! ) 1) + 1! + 1! = 1! ft Baroody Page of 11
Chapter 1 Test Review Question Answers. Find the volume and total surface area of the frustum shown Now, the small to the large by AA, so x x + = 1 x = 1 1 V Frustum = V Large Cone - V Smaller Cone = 1 πr )H Large Cone - 1 πr )H Small Cone = 1 1π) 1) - 1 π) 1) 1 = π - π = 0π u L large cone slant) = 1 + 1 = 1 l small cone slant) = 1 + = 1 A T.S. = πrl - πrl + πr + πr = π 1) 1) - π ) 1) + 1π + π = 0π 1 + 0π ft. Find the volume and the surface area. Measurements are in centimeters. 1 V = 1 )!r = 1 )!1 = 7! = 0! cm A T.S. =!r =! 1 ) = 7! cm Baroody Page of 11
Chapter 1 Test Review Question Answers 10. Find the volume and total surface area of a hemisphere with a 0 wedge cut out if the radius is inches. 0 V = 70 0 V Hemisphere) = 1 )) πr = 1 π ) = π in A TS = A LS + A B ) 1 ) πr + 1 ) πr = 1 ) + πr ) = π + π + π = π + π + π = 17π in 11. Find the radius of the base. The volume of the solid is 10π cm. V = 1 )) πr = πr 10π = πr ) 0 17 = r r = 1 cm Baroody Page of 11
Chapter 1 Test Review Question Answers 1. Find the volume. Measurements are in centimeters. 1 1 V = BH + BH 10 = ) 1) 10) + ) ) 10) = 100 cm 1. Find the radius of the base. The volume of the cone is 17! cm. V = 17! = 1 BH = 1!r )H 17! = 1!r ) r 17! = 1!r r = 17! 1! = 1 r = 1 cm Baroody Page of 11
Chapter 1 Test Review Question Answers 1. Find the volume and total surface area of the solid shown below. V = 00 0 V Cylinder) =!r )H in =! ) 1) = 0! in 1 in A T.S. = A Base ) + A Lateral Surface of Cylinder) + A Rectangles ) 0 = )!r +!rh) + bh) =! ) +! ) 1) + ) 1) = 1! + 0! + 7 = 7! + 7) in 1. Identify the surface of rotation generated in the diagram below and compute its total surface area and volume. The surface of rotation is a cylinder with a cylindrical hole through its middle 1 A T.S. = A T.S. of Cylinder - A Holes + A L.S. of Inside Cylinder = πr ) + πrh - πr ) + πrh = 1π) + π ) ) - π) + π ) ) = π + 0π - 1π + 0π = π u V = V Large Cylinder - V Small Cylinder = πr )H - πr )H = π ) 1 - ) = π) 7) = π u Baroody Page 7 of 11
Chapter 1 Test Review Question Answers 1. A right cylindrical log was cut parallel to the axis. Find the volume and the total surface area of the piece shown. 0 10 V = BH = A Segment ) H) = = ) - s [ )] H 0 0!r ) - 10 [ )] 0 1!) 10 ) ) 0 0! = [ - ] 0) 0 0 = 00! - 70 ) u A T.S. = A Segment ) + A Rectangle + 1 A L.S. of the Cylinder) ) + 10 = 0! - ) 0) + 1! ) 10) 0) = 100! - 0 + 00 + 100! 00! = - 0 + 00 ) u 17. A cylinder with diameter mm is cut twice at a angle as shown. Find the volume of this cylindrical solid. If you're interested I won't test you on this!), you can also find the surface area if you know that the ends of the "cannoli" are ellipses. V = V Cylinder - V Cutout Half Cylinder ) V = πr H 1-1 πr H ) = π 1) - π ) = 1π - π = 0π mm A TS = A LS + A B = 1 πrh Ends [ ) + πrh Center Cylinder ] + π r 1 r ), where r 1 is the semi-major axis and r is the semi-minor axis of the parabola forming the ends of the "cannoli." = π ) + π ) + π ) ) = π + π + 1π = 0π + 1π ) mm Baroody Page of 11
Chapter 1 Test Review Question Answers 1. Find the volume and surface area of a hexagonal prism with height 1 cm, and the length of one side of the regular hexagonal base being 10 cm. 1 V = BH [ )] H = s ) = [ ) ] 1) = 100 ) cm 10 [ )] + bh A T.S. = s ) = 00 + 10) 1) = 00 + 70) cm 1. A cube has a volume of 1 cm. A corner is cut off as shown in the diagram. Find the volume of this triangular pryamid and determine what fractional part of the cube it represents. V Cube = BH 1 = s )s = s H s = H = ) - ) = 1 - = 1 = V Pyramid = 1 BH = 1 ) ) = 1 = u ) V Pyramid V Cube = 1 = 1 Baroody Page of 11
Chapter 1 Test Review Question Answers 0. Each edge of the cube shown has a length of. The midpoint of each face of the cube is joined to form an eight-sided regular octahedron. Find the volume of the octahedron. Each edge of the regular octadedron measures as each is the hypotenuse of a --0 triangle with legs of length formed by connecting two face midpoints with the midpoint of an edge of the cube). Now, V = 1 BH ) = 1 ) H) = 1H...if only we knew the height! To find the height of half of the octahedron, we'll use the triangle shown. You should be able to figure out the two lengths shown below and then use the Pythagorean Theorem to find H or you can just recognize that it's since it's half the height of the cube!). H H = ) - ) = - 1 = = Show Stuff So, V = 1H = 1 ) = u 1. A "segment" of a grapefruit has diameter 1 cm and the flat faces form a 0 angle. Assuming the grapefruit was spherical, what are the volume and surface area of the segment? 1 V = 0 0 ) πr = 1 1 A TS = 1 1 πr ) + 1 πr ) = 7 1π π ) = π ) = 1π cm ) + π = π cm Baroody Page 10 of 11
Chapter 1 Test Review Question Answers. Find the volume and total surface area of the solid generated by rotating the figure below around the dashed line. All dimensions are in centimeters. 10 V = 1 πr H + πr H + 1 = 1 π ) πr ) ) ) + π ) 10 ) + 1 ) ) π = 1π + 10π + π = π + 10π) cm A TS = πrl + πrh + πr = π ) + π )10 + π ) = 1π + 0π + π = 7π + π) cm Baroody Page 11 of 11