document.write( "Question 1137649: The table shows the daily production level and profit for a business. Use the quadratic function \"+y=ax%5E2%2Bbx%2Bc+\"\r
\n" ); document.write( "\n" ); document.write( "to determine the number of units that should be produced each day for maximum profit. What is the maximum daily​ profit?\r
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\n" ); document.write( "x​ (Number of Units Produced​ Daily) | 30 | 50 | 100
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\n" ); document.write( "y​ (Daily Profit) | ​$5800 | $7320 | ​$4120\r
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\n" ); document.write( "\n" ); document.write( "Each day [??] units should be produced to have a maximum daily profit of ​$[????].
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Algebra.Com's Answer #755593 by MathLover1(20850)\"\" \"About 
You can put this solution on YOUR website!
\r
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\n" ); document.write( "\n" ); document.write( "The table shows the daily production level and profit for a business. Use the quadratic function \"+y=ax%5E2%2Bbx%2Bc+\"\r
\n" ); document.write( "\n" ); document.write( "to determine the number of units that should be produced each day for maximum profit. What is the maximum daily​ profit?\r
\n" ); document.write( "\n" ); document.write( "______________________________________________________________
\n" ); document.write( "\"x\"​ (Number of Units Produced​ Daily) | \"30\" | \"50+\" | \"100\"
\n" ); document.write( "______________________________________________________________
\n" ); document.write( "\"y\"​ (Daily Profit) | ​$\"5800\" | $\"7320\" | ​$\"4120\"\r
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\n" ); document.write( "\n" ); document.write( "\"+y=ax%5E2%2Bbx%2Bc+\"........plug in \"x=30\"​ and \"y=5800\"
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\n" ); document.write( "\"+5800=a%2A30%5E2%2Bb%2A30%2Bc+\"\r
\n" ); document.write( "\n" ); document.write( "\"+5800=900a%2B30b%2Bc+\"..........solve for \"c\"\r
\n" ); document.write( "\n" ); document.write( "\"+c=5800-900a-30b+\".......eq.1\r
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\n" ); document.write( "\n" ); document.write( "\"+y=ax%5E2%2Bbx%2Bc+\"........plug in \"x=50\"​ and \"y=7320\"
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\n" ); document.write( "\"+7320=a%2A50%5E2%2Bb%2A50%2Bc+\"\r
\n" ); document.write( "\n" ); document.write( "\"+7320=2500a%2B50b%2Bc+\"..........solve for \"c\"\r
\n" ); document.write( "\n" ); document.write( "\"+c=7320-2500a-50b+\".......eq.2\r
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\n" ); document.write( "\n" ); document.write( "\"+y=ax%5E2%2Bbx%2Bc+\"........plug in \"x=100\"​ and \"y=4120\"
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\n" ); document.write( "\"+4120=a%2A100%5E2%2Bb%2A100%2Bc+\"\r
\n" ); document.write( "\n" ); document.write( "\"+4120=10000a%2B100b%2Bc+\"..........solve for \"c\"\r
\n" ); document.write( "\n" ); document.write( "\"+c=4120-10000a-100b+\".......eq.3\r
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\n" ); document.write( "\n" ); document.write( "from eq.1 and eq.2 we have\r
\n" ); document.write( "\n" ); document.write( "\"5800-900a-30b+=7320-2500a-50b\"......solve for\"+b\"\r
\n" ); document.write( "\n" ); document.write( "\"50b-30b+=7320-2500a-5800%2B900a\"\r
\n" ); document.write( "\n" ); document.write( "\"20b+=1520-1600a\"\r
\n" ); document.write( "\n" ); document.write( "\"b+=76-80a\"...........eq.1a\r
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\n" ); document.write( "\n" ); document.write( "from eq.2 and eq.3 we have\r
\n" ); document.write( "\n" ); document.write( "\"7320-2500a-50b=4120-10000a-100b\" .....solve for \"b\"\r
\n" ); document.write( "\n" ); document.write( "\"100b-50b=4120-10000a-7320%2B2500a\" \r
\n" ); document.write( "\n" ); document.write( "\"50b=-7500a-3200\"\r
\n" ); document.write( "\n" ); document.write( "\"b=-150a-64\"..........eq2a\r
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\n" ); document.write( "\n" ); document.write( "from eq.1a and eq.2a we have\r
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\n" ); document.write( "\n" ); document.write( "\"76-80a=-150a-64\".....solve for\"+a\"\r
\n" ); document.write( "\n" ); document.write( "\"150a-80a=-76-64\"\r
\n" ); document.write( "\n" ); document.write( "\"70a=-140\"\r
\n" ); document.write( "\n" ); document.write( "\"a=-140%2F70\"\r
\n" ); document.write( "\n" ); document.write( "\"a=-2\"\r
\n" ); document.write( "\n" ); document.write( "go to \"b+=76-80a\"...........eq.1a, substitute \"a\"\r
\n" ); document.write( "\n" ); document.write( "\"b+=76-80%28-2%29\"\r
\n" ); document.write( "\n" ); document.write( "\"b+=76%2B160\"\r
\n" ); document.write( "\n" ); document.write( "\"b+=236\"\r
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\n" ); document.write( "\n" ); document.write( "go to \"+c=4120-10000a-100b+\".......eq.3, substitute \"a\" and \"b\"\r
\n" ); document.write( "\n" ); document.write( "\"+c=4120-10000%28-2%29-100%2A236+\"\r
\n" ); document.write( "\n" ); document.write( "\"+c=4120%2B20000-23600+\"\r
\n" ); document.write( "\n" ); document.write( "\"+c=4120-3600+\"\r
\n" ); document.write( "\n" ); document.write( "\"+c=520+\"\r
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\n" ); document.write( "\n" ); document.write( "so, your function is:\"+y=-2x%5E2%2B236x%2B520+\"\r
\n" ); document.write( "\n" ); document.write( "max is at vertex, so write your function in vertex form \"y=a%28x-h%29%5E2%2Bk\" where \"h\" and \"k\" are coordinates of the vertex \r
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\n" ); document.write( "\n" ); document.write( "\"+y=-2x%5E2%2B236x%2B520+\"....using completing square we have\r
\n" ); document.write( "\n" ); document.write( "\"+y=-2%28x%5E2-118x%29%2B520+\"\r
\n" ); document.write( "\n" ); document.write( "\"+y=-2%28x%5E2-118x%2Bb%5E2%29-%28-2b%5E2%29%2B520+\"....\"b=118%2F2=59\"\r
\n" ); document.write( "\n" ); document.write( "\"+y=-2%28x%5E2-118x%2B59%5E2%29%2B2%2A59%5E2%2B520+\"\r
\n" ); document.write( "\n" ); document.write( "\"+y=-2%28x-59%29%5E2%2B6962%2B520+\"\r
\n" ); document.write( "\n" ); document.write( "\"+y=-2%28x-59%29%5E2%2B7482+\"\r
\n" ); document.write( "\n" ); document.write( "=>\"h=59\" and \"k=7482\"\r
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\n" ); document.write( "\n" ); document.write( "Each day \"59\" units should be produced to have a maximum daily profit of ​$\"7482\".\r
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