SOLUTION: 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
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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:
F (p) = - 1/50 P^2 + 2P+ 20... 0 ≤ P≤ 100. Find the máximum- weight.
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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.
How do you find the vertex of a parabola? One way would be to rewrite the equation in vertex form:
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:
Since your equation is already in standard form we are going to use -b/2a:
which simplifies to:
p = 50 (which is in the allowed values of 0 to 100)
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):
which simplifies as follows:
The maximum weight then, is 70 grams (which occurred when the percent of yeast was 50).
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