The diagram shows three glasses (not drawn to scale). The measurements are all in centimeters.[img]http://s3.amazonaws.com/illustrativemathematics/images/000/000/488/medium/Task_1_23ab23a7105a270c99eeaa2a387bc727.jpg?1385557788[/img]The bowl of glass 1 is cylindrical. The inside diameter is 5 cm and the inside height is 6 cm.The bowl of glass 2 is composed of a hemisphere attached to cylinder. The inside diameter of both the hemisphere and the cylinder is 6 cm. The height of the cylinder is 3 cm.The bowl of glass 3 is an inverted cone. The inside diameter is 6 cm and the inside slant height is 6 cm.[list=1][*]Find the vertical height of the bowl of glass 3.[/*][*]Calculate the volume of the bowl of each of these glasses.[/*][*]Glass 2 is filled with water and then half the water is poured out. Find the height of the water.[/*][/list]
The solutions use volume formulas for cylinders, cones and spheres. Note that all units are in terms of centimeters: area units are cm2, and volume units are cm3. The units are sometimes omitted for convenience. The diagrams are not to scale.[list=1][*]To find the vertical height of the bowl portion of the glass, we use the Pythagorean Theorem.Height in centimeters=h=62−32−−−−−−√=36−9−−−−−√=27−−√. (Some may prefer to write 27−−√=33√.)[img]http://s3.amazonaws.com/illustrativemathematics/images/000/000/489/large/Sol_1_d5727ab3ffc315742b84a7114d0dd92d.jpg?1331699531[/img][/*][*][list=1][*]For glass 1 we use the fact that the volume of a cylinder is given by V=(area of base)⋅height. Since the base is a circle with radius r=52, the volume V (in cm3) is V=(π(52)2)⋅6=(752)π=3712π.[img]http://s3.amazonaws.com/illustrativemathematics/images/000/000/490/max/Sol_2_287136220bbb4b60b1914940bed61aba.jpg?1331699546[/img][/*][*]For glass 2 the bowl consists of 2 parts- a cylinder of height and radius 3 which sits atop a hemisphere of radius 3. We add the volumes of these to get the total volume. For the cylinder portion, the volume (in cm3) is (π32)⋅3=27π. For the volume of the hemisphere, take half the volume of a sphere of radius 3 to get: 12(43π33)=12(4⋅32π)=18π. Add these to get the total volume (in cm3): 27π+18π=45π.[img]http://s3.amazonaws.com/illustrativemathematics/images/000/000/491/max/Sol_3_0ace12cd7104b8e131edc3ba33e50b7c.jpg?1331699579[/img][/*][*]For the volume of the bowl of glass 3, use the fact that the volume of a cone is given by V=13Area of base⋅Height. The radius is 3and the height is 27−−√ by part (a). So the area of the base is 13π32and the volume V (in cm3 ) is V=13π3227−−√=327−−√π. (Or 93√π.)[img]http://s3.amazonaws.com/illustrativemathematics/images/000/000/492/max/Sol_4_19ed7b74efd529cd70f09af91ad78e54.jpg?1331699590[/img][/*][/list][/*][*]Note first that the height of the liquid is measured from the bottom of the bowl of the glass.The total volume of the glass is 45π, so if half the water is poured out then the remaining water occupies a volume of 452π , or 2212π. Notice that this is not just 3 cm up the glass since the hemispherical part of the glass holds less than the cylindrical part. (It is easy to see that if the glass had a cylindrical bottom, it would be bigger and so have a larger capacity.) As we saw in (b)(ii), the liquid in the glass fills the hemisphere first and then the cylindrical portion of the glass. The hemispherical part of the glass holds a volume of 18π, leaving a volume of 412π, or 92π, to fill the cylindrical portion.[img]http://s3.amazonaws.com/illustrativemathematics/images/000/000/493/max/Sol_5_cfb0e964dbee62e70f1834768d7ac1ff.jpg?1331699609[/img]The question is now reduced to finding the height of the liquid in the cylindrical portion of the glass. Once we know this height, we can simply add the height from the hemispherical portion, which is 3, and obtain the full height of the liquid in the glass.To get the height of the liquid in the cylinder we compute the height of a cylinder which has volume 92π. We use the formula for volume to solve for the height: 92π=π32h=9πh so h=12. We can now add 3 to this to get the required height of the liquid in the glass which is (in cm) 312.[img]http://s3.amazonaws.com/illustrativemathematics/images/000/000/494/max/Sol_6_e58465f543298ec350d704d7ee2c7cb2.jpg?1331699634[/img][/*][/list]