Question 633253
{{{F(p) = (- 1/50)P^2 + 2P+ 20}}}
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.<br>
How do you find the vertex of a parabola? One way would be to rewrite the equation in vertex form:
{{{4p(y-k) = (x-h)^2}}}
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:
{{{y = ax^2+bx+c}}}
Since your equation is already in standard form we are going to use -b/2a:
{{{p = -(2)/2(-1/50)}}}
which simplifies to:
p = 50 (which is in the allowed values of 0 to 100)<br>
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):
{{{f(50) = (-1/50)50^2 + 2(50) + 20}}}
which simplifies as follows:
{{{f(50) = (-1/50)2500 + 2(50) + 20}}}
{{{f(50) = -50 + 100 + 20}}}
{{{f(50) = 70}}}
The maximum weight then, is 70 grams (which occurred when the percent of yeast was 50).