John owns a hotdog stand. He has found that his profit is represented by the equation P(x)=x^2 + 12x + 46, where x is the number of hotdogs. What is the most he can earn?
Looks like he's in a good business. The more he sells, the more profit he makes, there's no limit. The minimum is $46 when he doesn't sell any hot dog. If he sells 10 hot-dogs, he makes $220 more!
Check the sign of the x^2 term!
To find the most profit John can earn, we need to find the maximum value of the given equation P(x) = x^2 + 12x + 46.
To determine the maximum value, we can use calculus or complete the square. Let's use the latter method:
1. Rewrite the equation in the form of a perfect square trinomial:
P(x) = (x^2 + 12x) + 46
2. To complete the square, we take half the coefficient of x (which is 12), square it (144/4 = 36), and add it to both sides of the equation:
P(x) + 36 = (x^2 + 12x + 36) + 46 + 36
P(x) + 36 = (x + 6)^2 + 82
3. Now, the equation is in the form P(x) + 36 = (x + 6)^2 + 82, and we can see that the maximum value occurs when (x + 6)^2 is equal to zero. This means x = -6.
4. Substituting x = -6 back into the original equation, we find the maximum profit:
P(-6) = (-6)^2 + 12(-6) + 46
P(-6) = 36 - 72 + 46
P(-6) = 10
Therefore, the most John can earn is $10 in profit.