A quadratic function word problem

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Question from liz, a student:

A kernel of popcorn contains water that expands when the kernel is heated,
causing it to pop. The equations below give the "popping volume"y(in cubic centimeters per gram)
of popcorn with moisture content x(as a percent of the popcorn's weight).

hot air popping: y = -0.761x^2 + 21.4x - 94.8
hot oil popping: y = -0.652x^2 + 17.7x -76.0

A) for hot air popping, what moisture content maximizes popping volume? What is the maximum volume?
B) For hot oil popping, what moisture content maximizes popping volume? What is the maximum volume?
C) the moisture content of popcorn typically ranges from 8% to 18%. Graph the equations for hot air and hot oil popping on the interval 8 less then or equal to x and less then or equal to 18.
D) Based on the graphs from part(c), what general statement can you make about the volume of popcorn produced from hot air popping versus hot oil popping for any moisture content in the interval8 less then or equal to x and less then or equal to 18.

Hi Liz,

I don't know how you are to determine the maximum value of these quadratics. For the quadratic y = ax² + bx + c with a negative, the maximum value occurs at x = -b/(2a). Maybe you were given this as a formula, maybe you know some calculus and can find this value using a derivative or you can use the symmetry of a parabola and notice that the maximum value occurs half way between the roots. In any case you can use this fact to solve parts A and B of your problem. For each of the expressions find y when x is 8% and 18% and use these values along with the values you found in parts A and B to sketch the graphs requires for part C.

I hope this helps,
Penny

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