document.write( "Question 192693This question is from textbook Intermediate Algebra
\n" ); document.write( ": Recall that total profit P is the difference between total revenue R and total cost C. Given R(x)=1000x-x^2 and C(x)=3000+20x, find the total profit, the maximum value of the total profit, and the value of x at which it occurs. \n" ); document.write( "
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Recall that total profit P is the difference between total revenue R and total cost C. Given R(x)=1000x-x^2 and C(x)=3000+20x, find the total profit, the maximum value of the total profit, and the value of x at which it occurs.
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\n" ); document.write( "Profit = Revenue - Cost
\n" ); document.write( "f(x) = R(x) - C(x)
\n" ); document.write( "f(x) = (1000x-x^2) - (3000+20x)
\n" ); document.write( "f(x) = 1000x-x^2 - 3000 - 20x
\n" ); document.write( "f(x) = -x^2 + 980x - 3000
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\n" ); document.write( "By looking at the coefficient associated with the x^2 term (-1), we know that it is a parabola that opens downward. Therefore, finding the vertex will provide the answer:
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\n" ); document.write( "To do this, manipulate the equation into the \"vertex form\":
\n" ); document.write( "y= a(x-h)^2+k
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\n" ); document.write( "f(x) = -x^2 + 980x - 3000
\n" ); document.write( "f(x) = (-x^2 + 980x) - 3000
\n" ); document.write( "f(x) = -(x^2 - 980x + __ ) - 3000
\n" ); document.write( "f(x) = -(x^2 - 980x + 240100) - 3000 + 240100
\n" ); document.write( "f(x) = -(x+490) + 237100
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\n" ); document.write( "maximum value of the total profit: $237100
\n" ); document.write( "value of x at which it occurs: -490
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