MATH 31 UNIT 5: LESSON 6A GEOMETRIC APPLICATIONS: MAXIMIZING AREA AND VOLUME. NAME ANSWERS Page 1 of 9
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1 NAME ANSWERS Page 1 of 9 EXAMPLE 1: The Starks have 60 metres of fencing with which to make a rectangular dog run. If they use a side of the shed as one side of the run, what dimensions will give the maimum area? shed P = w + l A = lw 60 = w + l l = 60 w A = w (60 w) A = 60w w da = 60 4w dt 0 = 60 4w w = 15 If w = 15 then l = 60 (15) = 0 The Starks should make their run 0 m by 15 m to get a maimum area of 450 m EXAMPLE : A rectangular field is to be enclosed and divided into smaller plots by a fence parallel to one of the sides. Find the dimensions of the largest such field if 100 m of fencing material is to be used. l P = w + l A = lw w w w 100 = w + l l = 100 w l = w A = w( w) = 600w 1.5w da = 600 w dt 0 = 600 w w = 00 The dimensions of the field should be 00 m by 00 m to get a maimum area of m U5 L6A Maimizing Area and Vol ANS
2 NAME ANSWERS Page of 9 EXAMPLE : A bo has square ends and the sides are congruent rectangles. The total area of the four sides and two ends is 94 cm. What are the dimensions of the bo if the volume is a maimum and what is the maimum volume? Surface area = + 4y volume = lwh 94 = + 4y V y y 94 4 dv d V V If = 7 then 94 (7) 7 4(7) The bo should be 7 cm by 7 cm by 7cm to produce a maimum volume of 4 cm U5 L6A Maimizing Area and Vol ANS
3 NAME ANSWERS Page of 9 EXAMPLE 4 An open (it has no top) bo has square ends and the sides are congruent rectangles. The total area of the four sides and one end is 19 cm. What are the dimensions of the bo if the volume is a maimum and what is the maimum volume? Surface area = + 4y volume = lwh y 19 = + 4y 19 4 dv d V V y V If = 8 then (8) The bo should be 8 cm by 8 cm by 4 cm to produce a maimum volume of 56 cm PRACTICE QUESTION 1. A storage bo has square ends and no top. The total surface area is 9600 cm. What are the dimensions of the bo if the volume is a maimum and what is the maimum volume? No top here U5 L6A Maimizing Area and Vol ANS
4 NAME ANSWERS Page 4 of 9 EXAMPLE 5 A page contains 600 cm. The margins at the top and bottom are cm. The margins at each side are to be cm. What are the dimensions of the paper if the printed area is a maimum? Dimensions of the printed area are ( 4) by (y 6) y Area of whole page A = y 600 = y Area of the print A = ( 4) (y 6) A = ( 4) ( 6) A = A = When = 0, The dimensions of the page are 0 cm by 0 cm U5 L6A Maimizing Area and Vol ANS
5 NAME ANSWERS Page 5 of 9 EXAMPLE 6 A rectangular open-topped bo is to be made from a piece of material 18 cm by 48 cm by cutting a square from each corner and turning up the sides. What size squares must be removed to maimize the capacity of the bo? 18 cm cm V = lwh= (18 )(48 ) = ( ) = V = = = 0 ( 18)( 4) = 0 = 18 or = 4 Too high The squares should be 4 cm by 4 cm to create a maimum volume of = 1600cm U5 L6A Maimizing Area and Vol ANS
6 NAME ANSWERS Page 6 of 9 ASSIGNMENT QUESTIONS 1. A carpenter is building a rectangular room with a fied perimeter of 54 feet. What are the dimensions of the largest room that can be built? What is its area? P = l + w 54 = l + w 7 = l + w l = 7 w A = lw A = w(7 w ) = 7w w A = 7 w 7 w = 0 w = 1.5 therefore l = = 1.5 The dimensions are 1.5 ft by 1.5 ft. and the maimum area is 18.5 square feet. A cattle farmer wants to build a rectangular fenced enclosure divided into five rectangular pens, as shown in the diagram. A total length of 10 m of fencing material is available. Find the overall dimensions of the enclosure that will make the total area a maimum. P = 6w + l 10 = 6w + l l = 10 6w l = 60 w A = lw A = w (60 w) = 60w w A = 60 6w 60 6w = 0 w = 10 l = 60 (10) = 0 The enclosure should be 10 m by 0 m to make the area a maimum U5 L6A Maimizing Area and Vol ANS
7 NAME ANSWERS Page 7 of 9. A bo has square ends, and the sides are congruent rectangles. The total area of the four sides and two ends is 16 cm. What are the dimensions of the bo if the volume is a maimum, and what is the maimum volume? Surface area = + 4y 16 = + 4y (6) If = 6 then 6 4(6) volume = lwh V V y V dv d The bo should be 6 cm by 6 cm by 6cm to produce a maimum volume of 16 cm 4. A storage bo has square ends and no top. The total surface area is 9600 cm. What are the dimensions of the bo if the volume is a maimum and what is the maimum volume? No top here The dimensions of the bo should be 40 cm by 5 ⅓ cm and the maimum volume will be ⅓ = 85 ⅓ cm S = + y 9600 = + y 9600 V ' V = y 9600 V 9600 V (40) U5 L6A Maimizing Area and Vol ANS
8 NAME ANSWERS Page 8 of 9 5. A rectangular page is to contain 150 cm of printing. The margins at the top and bottom of the page are cm. The margins at each side are cm. What should the dimensions of the page be if the minimum amount of paper is used? da d Dimensions of total page are + 4 by y + 6 Area of print: y = A ( 4) 6 A A The dimensions of the page should be = 14 cm by = 1 cm in order to use the minimum amount of paper. U5 L6A Maimizing Area and Vol ANS
9 NAME ANSWERS Page 9 of 9 6. From a thin piece of cardboard 0 cm by 0 cm, square corners are cut out so the sides can be folded up to make a bo. What dimensions will yield a bo of maimum volume? What is the maimum volume? V = (0 )(0 ) = 900 V = (0 )(0 ) V = V = = 1( ) 0 = ( 5)( 15) = 5 or = 15 0 (15) = 0 so = 15 is invalid The dimensions of the bo are 5 cm, by 0 cm by 0 cm. The maimum volume of the bo is 000 cm if the squares are 5 cm by 5 cm. U5 L6A Maimizing Area and Vol ANS
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