document.write( "Question 633223: Aki's bicycle designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by c(x)=0.2x²-1.3x+8.181, where c(x) is in hundreds of dollars.\r
\n" ); document.write( "\n" ); document.write( "The shop should build_____bicycles. \n" ); document.write( "

Algebra.Com's Answer #398848 by KMST(5399) $\"\"$ $\"About$

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$\"c%28x%29=0.2x%5E2-1.3x%2B8.181\"$ is a quadratic function.
\n" ); document.write( "As a quadratic function with a positive coefficient for the term in $\"x%5E2\"$ ,
\n" ); document.write( "it goes through a minimum.
\n" ); document.write( "Quadratic functions have the general form
\n" ); document.write( " $\"f%28x%29=ax%5E2%2Bbx%2Bc\"$
\n" ); document.write( "They graph as parabolas,
\n" ); document.write( "and have a minimum or maximum at $\"x=-b%2F2a\"$ .
\n" ); document.write( "In the case of $\"c%28x%29=0.2x%5E2-1.3x%2B8.181\"$ , there is a
\n" ); document.write( "minimum at $\"x=1.3%2F%282%2A0.2%29\"$ --> $\"x=1.3%2F0.4\"$ --> $\"highlight%28x=3.25%29\"$
\n" ); document.write( "So the cost per bicycle is minimum when $\"3.25\"$ hundred bicycles are built.
\n" ); document.write( "That is $\"highlight%28325%29bicycles\"$
\n" ); document.write( "
\n" ); document.write( "ABOUT QUADRATIC FUNCTIONS:
\n" ); document.write( "For a quadratic function, $\"f%28x%29=ax%5E2%2Bbx%2Bc\"$ ,
\n" ); document.write( "when the leading coefficient, $\"a\"$ , is positive,
\n" ); document.write( "the function grows without bounds as $\"x%5E2\"$ increases on both ends (positive and negative),
\n" ); document.write( "as the $\"ax%5E2\"$ term overwhelms whatever value the rest of the polynomial could take.
\n" ); document.write( "As a consequence the function looks like a smile, with a minimum in the middle:
\n" ); document.write( " $\"graph%28200%2C100%2C3%2C9%2C1%2C11%2C%28x-6%29%5E2%2B2%29\"$
\n" ); document.write( "If $\"a%3C0\"$ the shape of the graph is flipped and the function has a maximum:
\n" ); document.write( " $\"graph%28200%2C100%2C3%2C9%2C-12%2C-2%2C-%28x-6%29%5E2-3%29\"$
\n" ); document.write( "The maximum or minimum is the vertex of the parabola.
\n" ); document.write( "The function can be transformed algebraically from $\"f%28x%29=ax%5E2%2Bbx%2Bc\"$ to
\n" ); document.write( " $\"f%28x%29=a%28%28x%2Bb%2F2a%29%5E2-%28b%5E2-4ac%29%2F4a%5E2%29\"$
\n" ); document.write( "which shows that the extreme value (maximum or minimum) occurs when
\n" ); document.write( " $\"x%2Bb%2F2a=0\"$ , when $\"x=-b%2F2a\"$
\n" ); document.write( "and the line represented by $\"x=-b%2F2a\"$ is the axis of symmetry of the parabola that is the graph of the function.
\n" ); document.write( "If the expression reminds you of the quadratic formula, it is no coincidence.
\n" ); document.write( "The quadratic formula comes from the same
\n" ); document.write( " $\"f%28x%29=a%28%28x%2Bb%2F2a%29%5E2-%28b%5E2-4ac%29%2F4a%5E2%29\"$ making $\"f%28x%29=0\"$ . \n" ); document.write( "

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