CHAPTER 12 Circles and Circular Solids

Transcription

1 HAPTER 1 icles an icula Solis Section 1.1 icumfeences an Aeas of icles 1. Solution key: 1. heck stuents wok.. 3. π π 4. heck stuents wok. 5. heck stuents wok. 6. h ; b π. A bh ( ) Sample answe: The aea of the cicle is π. Sample ubic: Assign 1 point to coect wok in poblems 1, 4, an 5. Assign 1 point to each coect expession in poblems, 3, 6, an. Thee is a total of points. 3. (a) 3 6 The iamete is 6 metes. (b) 14 The aius is inches. 5. (a) (c) π ( 15 ) The cicumfeence is about 4.14 millimetes. π ( 9 ) The cicumfeence is about 8.9 inches. (b) The cicumfeence is about 34.5 feet. () The cicumfeence is about 5.14 metes. 015 engage Leaning. All Rights Reseve. May not be scanne, copie o uplicate, o poste to a publicly accessible website, in whole o in pat. 191

2 19 hapte 1 icles an icula Solis. (a) P π + 5 π (.5 ) The peimete of the semicicle is about 1.86 metes. (b) P ( 16.5 ) The peimete of the semicicle is about millimetes. (c) π P + 4 π ( ) The peimete of the semicicle is about 10.9 feet. () P ( 10 ) The peimete of the semicicle is about centimetes. 9. (a) L The length of the ac is 4 inches. (b) L πa π ( 40)( 9) 6.9 The length of the ac is about 6.9 metes. (c) L The length of the ac is about millimetes. () L πa π ( 10)( 6) 1.5 The length of the ac is about 1.5 centimetes. 11. (a) A 3.14( 1) The aea of the cicle is about squae feet. (b) The aius is 6 13 inches. A 3.14( 13) The aea of the cicle is about squae inches. (c) A π ( ) 154 The aea of the cicle is about 154 squae metes. () The aius is 3 16 centimetes. A 3.14( 16) The aea of the cicle is about squae centimetes. 015 engage Leaning. All Rights Reseve. May not be scanne, copie o uplicate, o poste to a publicly accessible website, in whole o in pat.

3 Section 1.1 icumfeences an Aeas of icles (a) (b) A π 135 π 5.5 ( ) The aea of the secto is about 5.5 squae centimetes. A π 11 π ( 9) 9.13 The aea of the secto is about 9.13 squae inches. (c) The aius is 1 6 inches. A π 55 π ( 6) 1. The aea of the secto is about 1. squae inches. () The aius is 8 4 metes. A π 63 π ( 4) 8.9 The aea of the secto is about 8.9 squae metes. 1. (a) The figue is compose of one semicicle an a tapezoi. icumfeence of semicicle: π 3.14() Peimete of tapezoi (not incluing top base of tapezoi): P So, the total peimete is about millimetes. (b) The figue is compose of fou equal semicicles. icumfeence of fou semicicles: π 4 ( 3.14)( 1) 6.8 So, the total peimete is about 6.8 metes. 15. (a) The figue is compose of two semicicles of equal aius an a paallelogam. The aius of each semicicle is 1 mete. Aea of two semicicles: π π A + () 1 π 3.14 Aea of paallelogam: A ( 3.5)( 1.5) 5.5 So, the total aea is about squae metes. (b) The figue is compose of two of the same sectos, an equilateal tiangle, an an isosceles tiangle. Aea of two equal sectos: A π π ( ) Let x be the height of the equilateal tiangle. Then, the height of the isosceles tiangle is 6 x. Aea of two tiangles: 1 1 A ( x)( 4.5) + ( 6 x)( 4.5) 1 ( 4.5 )( x + 6 x) 1 ( 4.5 )( 6 ) 13.5 So, the total aea is about squae inches. 015 engage Leaning. All Rights Reseve. May not be scanne, copie o uplicate, o poste to a publicly accessible website, in whole o in pat.

4 194 hapte 1 icles an icula Solis 19. π So, the cicumfeence of the pizza is about 38 inches. 1. A π ( ) π ( 4) 50.4 So, the aea of the logo is about 50 squae yas. 3. (a) Peimete of e squae: About 9 mm Peimete of blue squae: About 64 mm Diamete of cicle: About 3 mm (b) Ratio of peimete of e squae to iamete of cicle: Ratio of peimete of blue squae to iamete of cicle: (c) Aveage: () Sample answe: Use egula polygons with moe sies to pouce bette appoximations of π. 5. Since 1 mile 63,360 inches, then miles 443,50 inches. π 8π 8 88 The cicumfeence of the tie is about 88 inches. Since 443, , the tie will make about 5040 otations.. The angle between any two consecutive pouns is So, when the neele is at the -poun 10 mak, the angle between the pevious position an the new position of the neele is 36. The length of the ac is given by: πa L π ( )( 3) Theefoe, the tip of the neele has move about 3. inches. 9. (a) The aius of each semicicle is metes fo the fist lane, an about metes fo the secon lane. So, the length of each semicicula ac fo the fist lane is given by: ( 36.5) L an fo the secon lane is given by: ( 3.) L Theefoe, the istance aoun the tack along the insie of the fist lane is about ( ) metes, an the istance aoun the tack along the insie of the secon lane is about ( ) metes. The istance between the stating blocks in metes. (b) Sample answe: Since the unne in the innemost lane will un a shote istance than the unne in the next lane, the stating blocks ae staggee so that evey unne coves the same istance uing the ace. 31. (a) 1. Unestan the Poblem You nee to fin the aea of the egion that is shae.. Make a Plan Fin the aea of the lage cicle, an then subtact the total aeas of the two smalle cicles fom it. 3. Solve the Poblem The aius of each of the small cicles is inches, an the aius of the lage cicle is 14 inches. So, the aea of the two smalle cicles is π π 30. an the aea of () () the lage cicle is π( ) π( ) Theefoe, the aea of the shae egion is about squae inches. (b) 1. Unestan the Poblem You nee to fin the aea of the egion that is shae.. Make a Plan Fist, subtact the aea of a semicicle fom half the aea of the squae. This will give the aea of one of the fou non-shae egions between the petal shapes. So, the equie aea is obtaine by subtacting the fou non-shae aeas fom the total aea of the squae. 3. Solve the Poblem The aius of a semicicle is 4 centimetes. So, the aea of the π() π( ) semicicle is 6.8 an the aea of half of the squae is Theefoe, the aea of one of the non-shae egions is about squae centimetes. The aea of the shae egion is about 4 4( 1.) 9.1 squae centimetes. 015 engage Leaning. All Rights Reseve. May not be scanne, copie o uplicate, o poste to a publicly accessible website, in whole o in pat.

5 Section 1. Suface Aeas of icula Solis Unestan the Poblem You nee to fin the length of the cuve that sepaates the two egions an the aea of the shae egion.. Make a Plan To calculate the length of the cuve, you nee to fin the cicumfeence of the smalle cicle fome by the two equal semicicles. Because the two semicicles ae of equal aeas, you can fit the shae semicicle in the place of the non-shae one. Theefoe, the aea of the shae egion can be calculate by fining the aea of half of the lage cicle. 3. Solve the Poblem The aius of each of the semicicles is 6 3millimetes, an the aius of the lage cicle is 6 millimetes. The cicumfeence of the smalle cicle is () So, the length of the cuve is about π() π( 6) millimetes. The aea of half of the lage cicle is Theefoe, the aea of the shae egion is about 56.5 squae millimetes. 35. Sample answe: The stuent use the iamete instea of the aius, an shoul eview the efinition of a cicle. 3. (a) Sample answe: Fom the two iagams, it seems that Option A will iigate moe lan since the total aea enclose by one lage cicle seems to be geate than the total aea enclose by fou smalle cicles in Option B. (b) Let be the length of a sie of the squae. Then the aea of the squae is. The aea of the cicle is π pecentage of the fiel that will be iigate is %. (c) Let be the length of a sie of the squae. Then the aea of the squae is. The aea of each cicle is π. So, the π. So, the 4 4 π 4 pecentage of the fiel that will be iigate is %. () Sample answe: The answe in pat (a) oes not match with pats (b) an (c). Although fom the two iagams it seems that Option A will iigate moe lan than Option B, but that is not the case. The aea covee by one lage cicle in Option A is the same as the aea covee by fou smalle cicles in Option B. Theefoe, both options will iigate the same amount of lan. 39. Yes. Sample answe: If is the aius of a cicle, then the atio of the cicumfeence to the aius is π : π : 1, an theefoe is the same fo evey cicle. Section 1. Suface Aeas of icula Solis 1. Solution key: 1. Sample answe: 3 cm; The cicles ae conguent.. Sample answe: w 5 cm, cm Because the ectangula piece of pape is wappe aoun the can, the length of the ectangle is the same as the cicumfeence of the base of the can S Aea of two cicles + Aea of ectangle + h 5. Sample answe: () ()() S cm Sample ubic: Assign 1 point to each coect quantity, elationship, fomula, o net. Thee is a total of 8 points. 015 engage Leaning. All Rights Reseve. May not be scanne, copie o uplicate, o poste to a publicly accessible website, in whole o in pat.