document.write( "Question 633253: there were studied the effects on the rats fed on a diet that contains 10 % of protein. The protein was consisting in of yeast and cornflour. changing the percentage P of yeast in the miscellany of the above mentioned protein, it was estimated that the weight gained (in grams) of a rat in a period of time, it was f (p), where:
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\n" ); document.write( " F (p) = - 1/50 P^2 + 2P+ 20... 0 ≤ P≤ 100. Find the máximum- weight.
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Algebra.Com's Answer #398863 by jsmallt9(3758)\"\" \"About 
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\"F%28p%29+=+%28-+1%2F50%29P%5E2+%2B+2P%2B+20\"
\n" ); document.write( "As you probably know, the graph of f(p) will be a parabola because of the p squared term. And since the p squared term has a negative coefficient, -1/50, this parabola will open downwards. If you picture such a parabola in your mind (or draw one) it should be easy to see that there is a highest (maximum) point on this parabola, the vertex. So if we find the vertex we will be finding the maximum weight.

\n" ); document.write( "How do you find the vertex of a parabola? One way would be to rewrite the equation in vertex form:
\n" ); document.write( "\"4p%28y-k%29+=+%28x-h%29%5E2\"
\n" ); document.write( "Another way to find the vertex is to know and remember that the x-coordinate (or p-coordinate in this case) of the vertex is -b/2a with the \"b\" and \"a\" coming from the standard form of the equation of a parabola:
\n" ); document.write( "\"y+=+ax%5E2%2Bbx%2Bc\"
\n" ); document.write( "Since your equation is already in standard form we are going to use -b/2a:
\n" ); document.write( "\"p+=+-%282%29%2F2%28-1%2F50%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "p = 50 (which is in the allowed values of 0 to 100)

\n" ); document.write( "Of course we have to remember that p is the percent of of yeast, not the maximum weight. To find the maximum wedight we must find f(50):
\n" ); document.write( "\"f%2850%29+=+%28-1%2F50%2950%5E2+%2B+2%2850%29+%2B+20\"
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "\"f%2850%29+=+%28-1%2F50%292500+%2B+2%2850%29+%2B+20\"
\n" ); document.write( "\"f%2850%29+=+-50+%2B+100+%2B+20\"
\n" ); document.write( "\"f%2850%29+=+70\"
\n" ); document.write( "The maximum weight then, is 70 grams (which occurred when the percent of yeast was 50).
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