Kg hg dag g dg cg mg. Km hm dam m dm cm mm

Transcription

1 Metric System Conversions Mass (g = gram) Kg hg dag g dg cg mg Distance or Length ( m = metre) Km hm dam m dm cm mm Area (m = square metre) Km hm dam m dm cm mm Volume (m = cubic metre) Km hm dam m dm cm mm Capacity (l = litre) kl hl dal l dl cl ml volume vs capacity m = kl dm = l cm = ml

2 8/0/0 6.- A- Areas of Prisms and Cylinders For every solid we can find two types of areas: Lateral area (A L ): area of lateral faces-no base(s) Total area (A T ): area of lateral faces and base(s) Formulas: A L (Lateral Area) A T = A L + A b (Area total) Cubes w 6w Prisms P b h P b h + A b Cylinders or + Ex : A T = 6w = 6() = cm cm Ex : Page 78 # ) cm cm? 8 cm -- pythagoreantriple A T = P b h+ A b = (++)(8) + ( ) = ()(8) + (6) = 96 + = 08 cm Ex : P.8 # b) If d =. cm Then r =.7 cm 8. cm A T = πrh + πr = π(.7)(8.) + π(.7) = π(.7)(8.) + π(7.9) =.6π+.8π = 9.9πcm = 88.6 cm page 78 #,8,9,0 page 8 # Ex : Find the total surface area of this prism A T = P b h + A b = (++)() + ( ) = + (. ) = +.6 = 6.6 cm cm cm Find the height h a = c b a = (cm) (.cm) a = 8.7cm a. cm 6 Ex : Find the total surface area of the cylinder r= cm h=0 cm A T = πrh + πr = π()(0) + π() = 00π+ 0π = 0πcm Do and Hand in: page 79 #, page 8 #,6,7 7 8

3 8/0/0 6.- B- Pyramids Ex : Find the total surface area of the pyramid of Giza. S=86. m Formula: Pyramid A L (Lateral Area) A T = A L + A b (Area total) + A b A T = + A b =.. + (0.) = L base =0. m A T =7 7.9 m Ex : P. 8 # 8c) 8 cm cm A L = = = cm Ex : P. 8 # 0.8 m 0.6 m.m 0. m A upperbase = (0.6) = 0.6 m A L Block = A L Big pyr A L Small pyr = - = =.6. =. m Total area for 000 Blocks = 80 m A total = Number of cans = 80 / 0 =.8 m = 7 cans page 8 # 8, 0,,,,

4 8/0/0 6. C- Cones Ex : Find the total surface area of the cone. Formula: A L (Lateral Area) A T = A L + A b (Area total) Cone π π + πr.6 cm 0. cm S =.6 0. =.67 cm A T = π + πr = π(0.)(.67) + π(0.) =.π+ 0. π =.8πcm A T.0 cm cm Ex : page 8 # 6 AT = A Lwhole A Ltop + A base + A top = cm cm cm = π()(6) -π()() + π() + π() = 8 π-π+ 9π+ π = 6 πcm or 8.6 cm page 8 # 6 to 9

5 /06/07 Formulas: 6. D- Spheres A L (Lateral Area) A T = A L + A b (Only hemispheres have a base) Spheres Hemispheres Ex : A Tennis ball has diameter 6 cm. What is its surface area? A total = = π ( ) = 6 π =.0 cm 6 cm Ex : A soccer ball with diameter 0 cm is placed tightly inside a cube box. Find the difference between their surface areas. A = A cube -A sphere = 6w - = 6(0) - = 6(900) -900π 0 cm = = 7.7 cm Ex : How many times greater is the SA of the Sun than the Earth? Ratio = A Sun A Sun A Earth = = π (6900 ) =.99x0 π = x0 km A Earth = = π (678 ) =.67x0 8 π =.9x0 8 km d S = km r = 6900 d E = 76 km r = 678 = 6.08x0 km.x0 8 km = 89 So the sun has a SA 89 x bigger than the earth s page 8 # 7 to

6 /06/07 6. Areas of Decomposable solids Unfamiliar Solids can be broken down into simpler solids so that their area and volume can be calculated more easily. When you have a decomposable solid:. Separate and identify the solids involved.. Write the formulas for all the areas involved.. Calculate them and add them together.. Watch out for hidden bases whose areas should not be included. SOLIDS BASIC FORMULAS LATERAL AREA TOTAL AREA RIGHT PRISMS A LAT = P B h A TOT = A LAT + A B RIGHT CYLINDERS A LAT = πrh = πdh RIGHT REGULAR PYRAMIDS A LAT = A TOT = πrh+ πr = πdh + πr A TOT = A LAT + A b RIGHT CONES A LAT = πrs A TOT = πrs+ πr SPHERES A LAT = A TOT = πr HEMISPHERE A LAT = πr A TOT = πr Ex : Activity page 87. Cone: r = ; h = ; s=? s = + = + = =. Cylinder: r = ; h = 8. Hemisphere: r = cm Total S.A = A L CONE + A L CYL. + A L HEMI. = πrs + πrh + πr cm = π()() + π()(8) + π( ) Note the hidden bases = π + 8π + 8π between the cone, = 8πcm cylinder, and the hemisphere. or.7 cm cm 8 cm Ex : The solid shown below consists of a rectangular base pyramid joined to a rectangular base prism. The height of each lateral surface of the pyramid is cm, and the prism s dimensions are 0 cm by 8 cm and a height of 6 cm. Find the total surface area. Note the hidden base between the pyramid and the prism. A t = A Lpyr. + A Lprism + A b = + h + lw = ()() + +()() = = cm Ex : p. 89 # 8 A hemisphere is placed on the flat surface of another hemisphere. What is, rounded to the nearest unit, the total area of this solid? cm h r A T = A LHem + A Lhem + A B A b = πr + πr + πr πr = π(6) + π() + π(6) - π() = 7 π+ 8 π+ 6 π-9π = 7 πcm Or = 68 cm H 6 cm R 6 page 88 # to 6, 7

7 9/0/0 6. Volumes of Solids Volumeis the quantity of space that an object occupies Capacityis the volume of the hollow part of an object. A) The main unit of volume is the cubic meter (m ). It corresponds to the space enclosed in a cube with edge m. Length (m) m Area (m ) m Volume (m ) B) The cubic decimetre (dm ) corresponds to a cube with edge dm(0 cm) L dm = litre L of water = kg at standard temperature and pressure C) m = KL dm = L cm = ml Examples Convert each of the following: dm = cm cm = m dam = dm m = hm =.69 x hm = mm Convert each of the following: m = L dm = L L = dm KL = dm ml = mm cm = dm = L = ml page 90 # ; page 9 # 9. 6

8 9/0/0 6.-B-Volume of a Prism Consider a pack of printer papers. With dimensions: l = 8 cm; w =. cm; h = cm. Area of one paper: A = lw= (8)(.) = 60 cm The space this pack occupies (volume): = (60)() = 00 cm If we shift the pack a little, the # of papers (height) stays the same and so does the area of each paper. Therefore the volume stays the same too. Recall the AREAof a: V prism = A b h. Rectangle = l w. Square = b. Parallelogram = b h. Circle= πr. Triangle = 7. Rhombus = 6. Trapezoid = ( ) h 8. Regular polygon = Ex : Determine the volume of this cube Ex : Determine the volume of this prism V cube = A b h = w w V cube = w = = 7 cm cm V prism = A b h 8. dm = h = (.)(.) () = dm. dm height = dm Ex : Page 9 #9 This tank is filled with 00 L of gas. How much time will be required to fill the rest of it at a rate of 0 L/min? V tank = A b h = ()()() = 6 m = 6 KL = 6000 L V emptyspace = = 00 L Minutes required = 00/0 = min m m m page 9 # 7,,, 6, 8, 0 6

9 9/0/0 Ex : Determine the volume of this cylinder V = πr h 6.-C- Volume of a Cylinder V cylinder = πr h = π( )() = π = 0.0 cm cm cm 7 V cylinder = πr h Ex : Find the volume of this hot air balloon D = m R = 7 m H = m V = πr h = π(7) () = 7π = 8. m Diameter = m Height is. times the diameter 8 Ex : Determine the volume of material required to make a cd in ml Thickness is mm 8 mm 0 mm r = 9 mm R = 60 mm h = mm V =πr h - πr h =π(60) () π(9) () =00π π = 96π = mm = 0.68 cm = 0.68 ml 9 Ex : page 96 # A triangular base prism is submerged in a cylindrical bucket of water with a cm radius. The water level in the bucket rises 0.8 cm. What is the volume of the prism cm submerged in the water? V prism = V increase in height = πr h = π() (0.8) = 0π cm = 6.8 cm 0.8 cm 0 Worksheet + page 9 #, 7, 9

10 9/0/0 6.-D-Volumes of Pyramids and Cones The volume of any pyramidor cone is / that of a prism of the same height and base. V pyramid = V = V cone = V pyramid = Ex : Find the volume m m m V pyramid = = ()()() = 8 m V cone = O 6cm V cone = cm = ( )() = 0πcm Ex p. 98 # 9 A square base pyramid trophy, has height 6 cm and its slant height is 0 cm. If it is made of aluminum and the mass of dm of aluminum is.7 Kg, calculate the mass of this trophy. V pyramid = 6 cm 0 cm 0.6 dm dm = (.)(.)(.) a = 0. dm a = 0.6 = 0.8 dm Base =.6 dm Mass = (0.)(.7) =.8 Kg Ex : p. 99 # 8 If Eric fills the plastic cup (shown below) to its height with lemonade, how much lemonade (in cl). cm will be poured into the cup? V cup = V bigcone V bottom cone = - = (.) (.) - (.) cm =.π-.π = 0.π V lemonade= (7.78) = 7.78 cm = 8.9 cm = 8.9 ml =.8 cl 9 cm.6 cm page 98 #,7, page 99 #,6,

11 9/0/0 6.-E- Volume of a Sphere V sphere = V sphere = V cone = ( ) = ( ) = r V sphere = r r r Ex : Find the volume of the sphere with diameter 0 cm V sphere = = ( ) =. cm 0 cm Ex : Find the volume of the Earth in km r = 6 78 km diameter= 76 km V earth = = ( ) =.09 x0 km Ex : p.0 # 7 The interior and exterior diameters of a metallic ball are 8 cm and 0 cm respectively. What is the ball s mass if the mass of dm of this metal is Kg? V metal = V outside V inside = - = ( ) - ( ) = =.86 cm = 0. dm Mass = 0.() = Kg 0 cm 8 cm page 00 # 9(a,b),6,68,70,7

12 9/0/0 6. Volume of decomposable solids. Separate and identify the solids involved.. Write the formulas for all the areas involved.. Calculate them and add them together. Ex : Find the volume of this silo in Kl: V silo = V cylinder. + V hemisphere = π r h + = π (.) (6) + (. ) =. π +. π =.7 π m = 9.6 m = 9.6 kl m 6m Ex : Find the volume of this ice cream cone in ml: Ex : Determine the volume of this space probe V icecream cone = V cone + V hemisphere = + = (.) () + (.) =.8 π + 0. π = π cm = 78. cm = 78. ml cm 7 cm V silo =V cone + V cylinder. + V hemisphere = + π r h + = () () + π () (6) + ( ) = π + π + 8 π = 96 π m = 0. m SOLIDS LATERAL AREA TOTAL AREA Volume RIGHT PRISMS A LAT = P B h A TOT = P B h + A B V prism = A b h RIGHT CYLINDERS A LAT = πrh A TOT = πrh+ πr V cylinder = πr h page 0 #,,7,8 RIGHT REGULAR PYRAMIDS A LAT = A TOT = + A b V pyramid = RIGHT CONES A LAT = πrs A TOT = πrs+ πr V cone = SPHERES A LAT = A TOT = πr V sphere = HEMISPHERE A LAT = A TOT = πr Note: if the base is included, add πr V sphere= 6

13 /06/07 6. Missing measures of a solid Sometimes we know the Volume or Surface Area of an object and we need to find the, radius height or side length. When looking for a missing measure of a solid:. choose and write down the applicable formula based on the information given. substitute in all known values. solve for the unknown value Ex : Find the radius of the sphere V=08π cm V = πr πr 08π = 608π = πr 608 r = = r r = r=. cm Ex : Find the height of the cylinder A T = πrh + πr 6.9 = π(.)h + π(.) 6.9 = 7.h = 7.h. = h So h =. m A T =6.9 m r=. m Ex : Find the side length of the cube Ex : Find the radius of the cylinder Ex : Find the side length of the cube V=8. cm V = w 8.= w w = 8. w= 8. w =.7 cm V= 8 cm h= cm V =πr h 8=πr =π r π π = r r = r= 8 cm A T = 6w 9 = 6w 9 = w w = 7 cm A T =9 cm 6 page 0-08 #, 7, 8,, 7, 8, 7

14 SOLIDS LATERAL AREA TOTAL AREA Volume RIGHT PRISMS A LAT = P B h A TOT = P B h + A B V prism = A b h RIGHT CYLINDERS A LAT = πrh A TOT = πrh+ πr V cylinder = πr h RIGHT REGULAR PYRAMIDS A LAT = A TOT = + A b V pyramid = RIGHT CONES A LAT = πrs A TOT = πrs+ πr V cone = SPHERES A LAT = A TOT = πr V sphere = HEMISPHERE A LAT = A TOT = πr Note: if the base is included, add πr V sphere=