Geometry Problem Solving and Volume Calculations
Solve challenging geometry problems involving spheres, cones, cylinders, and frustums. Calculate volumes, heights, and surface areas of various geometric shapes using proportional relationships and volume formulas. Explore concepts of mass, density, and similarity in geometric figures. Test your problem-solving skills with calculations and comparisons of different shapes and volumes.
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On your whiteboards A : B 5 : 6 25 : 36 125 : 216 LSF ASF VSF Volume of B = 5000 125 x 216 = 8640 cm3 X : Y 2 : 3 4 : 9 8 : 27 LSF ASF VSF SA of X = 225 9 x 4 = 100 cm2
Small : Large 1 : 1.44 1 : 1.2 ASF LSF Height 10cm : 12cm (44% increase) (10cm x 1.2)
Did you expect these answers? Small : Large 1/5 : 1 1/125 : 1 LSF VSF Mass is directly proportional to volume. So the mass will increase by the same factor as the volume if the two spheres have the same density. Mass of smaller spheres = 375 x 1/125 = 3kg (1/5 of the original) (1/5 cubed) Small : Large 1/5 : 1 1/25 : 1 LSF ASF SA of 1 smaller sphere = 8000 x 1/25 = 320 cm2 (1/5 of the original) (1/5 squared) If each small sphere has a volume 1/125 of the original, then there must be 125 smaller spheres. Thus, the total SA = 125 x 320 = 40,000 cm2.
If the total height is 3h, then the height of the tip of the cone (which is similar to the whole cone) is 1h. LSF VSF Tip of Cone : Whole Cone 1 : 3 1 : 27 The volume of the whole cone is 27 times bigger than the tip. Volume of small cone Volume of cup (a frustum ) = 459 17 = 442cm3 = 459cm3 27 = 17cm3
Challenge Total volume of all 3 parts = 125 All parts have same volume = 125 4? 4? 3 = 125 12? =2 3??3 = 125 Volume of Hemisphere 12? 2 3?3 = 125 12 ?3 = 375 24 = 125 8 ? = 5 2
Volume of cylinder: Challenge ??2? =125 12? ?2? =125 12 2 5 2 ? =125 12 2 All parts have same volume = 125 4? 3 = 125 ? =125 5 2 =5 12? 12 3 Length of the hemisphere is the radius = 5 Volume of cone: 2 Length of the cylinder is the height = 5 3 The volume of a cone is 1/3 of the volume of a cylinder with the same base and height. = 5 Length of the cone is the height hemisphere : cylinder : cone : 15 : 3 : If a cone and cylinder with the same base also have the same volume, then the height must be 3 times bigger. Check it! 5 2 5 3 : 5 10 : 30 2 : 6 (x6) ( 5)
Challenge Length of the hemisphere is the radius = 5 2 cm 3cm Length of the cylinder is the height = 5 Length of the cone is the height = 5 cm Total length = 5 2+5 3+ 5 =55 6= 9.1 6 ??
