Q:
John owns a hotdog stand. He has found that his profit is represented by the function p(x)=-x^2+66x+81, with p(x) being profits and x the number of hotdogs sold. How many hotdogs must he sell to earn the most profit.
my answer would be a million bajillion hot dogs lol. but my answer key says 33. help please
A:
haha yes, it would be nice if he could sell a million bajillion hot dogs, but look at your equation, do you notice that -x^2 term? that means that when he sells a lot of hot dogs, he'll be subtracting that huge number squared from his profits! that's really really bad for his profits!
also, if you'll notice, the formula for his profits is the equation of a parabola! not just any parabola, but one that is opening downwards. this means that there is maximum height that his parabola reaches, corresponding to his maximum profit, which makes this a maximization/minimization problem!
now, all that analysis aside, here's how you solve it:
since the slope (derivative, in other words) of his profits curve (the parabola) is zero at its maximim, we can take the dervative of the profits curve and set that equal to zero to solve the number of hot dogs:
P(x) = -x^2 +66x +81
so the derivative, P'(x) is
P'(x) = -2x +66
now we set that equal to zero (for the reasons mentioned above),
0 = -2x + 66
and solve for x (the number of hot dogs),
2x = 66
x= 33
voila!
to solve these problems in the future, try to analyze the problem as i did in the first two paragraphs. this will really help you get a grasp of the mathematics, and make solving these kinds of problems a breeze
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