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3 The formula for the area of a triangle is A 1 BH The area is 20 2 B H 14 so H 14 B The formula becomes 20 1 B 14 B 2 1 20 B 14 B 2 40 B 14 B 40 14B B2 0 14B B2 40 0 14B B2 40 0 14B B2 40 0 B2 14B 40 0 B 10 B 4 B 10 0 B 4 0 B 10 B 4 There are two triangles that satisfy the conditions If the base is 10 inches, the height is 14 10 4 inches If the base is 4 inches, the height is 14 4 10 inches 4 By the Pythagorean theorem, a2 b2 c2 The hypotenuse is c, so c 85 One leg is 71 longer than the other so a b 71 (b a 71 also works) The Pythagorean theorem becomes 2 2 2 85 b 71 b 852 b 71 2 b2 7225 b 71 b 71 b2 7225 b2 142b 5041 b2 7225 2b2 142b 5041 0 2b2 142b 2184 1 1 0 2b2 142b 2184 2 2 0 b2 71b 1092 0 b 13 b 84
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b 13 0 b 13
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b 84 0 (This does not lead to a solution)
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The shorter leg is 13 cm and the longer leg is 13 71 84 cm 5 Because the can s diameter is 6, the radius is 3 Let x represent the increase in the can s radius The radius of the new can is 3 x The volume of the current can is V r2 h  3 2 5 45: To increase the volume by 50% means to add half of 45 to itself; the new 1 90 45 135 volume would be 45 45 The volume 2 2 2 2 135 formula for the new can becomes  3 x 2 5 2 135  3 x 2 5 2 1 135 3 x 2 5 2 27 3 x 2 2 27 3 x 3 x 2 27 9 6x x2 2   27 2 2 9 6x x2 2 27 18 12x 2x2 27 2x2 12x 18 0 2x2 12x 9 p p 122 4 2 9 12 144 72 x 4 2 2 p p p 2 12 216 12 6 6 12 6 6 4 p 4p 4 2 6 3 6 6 3 6 % 0:674: 2 4 12 (The other solution is negative) The manufacturer should increase the can s radius by about 0674 inches Because the diameter is twice the radius, the manufacturer
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should increase the can s diameter by about 2 0:674 1:348 inches 6 A pizza s shape is circular so we need the area formula for a circle which is A r2 The radius is half the diameter, so the restaurant s large pizza has a radius of 8 inches The area of the restaurant s large pizza is  8 2 64 % 201 The restaurant s large pizza is 20% larger than the competition s large pizza Let A represent the area of the competition s large pizza Then 201 is 20% more than A: 201 A 0:20A A 1 0:20 1:20A 201 1:20A 201 A 1:20 167:5 A A r2 167:5 r2 167:5 r2  r 167:5 r  7:3 % r The competition s radius is approximately 73 inches, so its diameter is approximately 2 7:3 14:6 inches
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Distance Problems
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There are several distance problems that quadratic equations can solve One of these types is stream problems: a vehicle travels the same distance up and back where in one direction, the stream s average speed is added to the vehicle s speed and in the other, the stream s average speed is subtracted from the vehicle s speed Another type involves two bodies moving away from each other where their paths form a right angle (for instance, one travels north and the other west) Finally, the last type is where a vehicle makes a round trip that takes longer in one direction than in the other In all of these types, the formula D RT is key
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