Charlie sells beaded bracelets as a side job. Charlie will have a maximum profit of $12 when he sells 5 bracelets. Construct Charlie’s profit function, p(x), to model his sales, in dollars, as a a function of the number of bracelets he sells.
p(x) = [a] (x-[h])^2 + [k]
Find [k]
Given that Charlie will have a maximum profit of $12 when he sells 5 bracelets, we can plug in these values into the profit function p(x) and solve for [k].
p(5) = 12
12 = [a] (5 - [h])^2 + [k]
Since the profit function p(x) is a quadratic function, we can rewrite it in standard form:
p(x) = ax^2 + bx + c
Plugging in the given values, we get:
12 = a(5 - [h])^2 + [k]
Expand and simplify:
12 = a(25 - 10[h] + [h]^2) + [k]
12 = 25a - 10ah + ah^2 + [k]
Now we can set up a system of equations:
25a = [k] (1)
-10ah = 0 (2)
ah^2 = 12 (3)
From equation (2), we know that either a = 0 or h = 0, but since we are dealing with a profit function, a cannot be zero. So, h = 0 from equation (2).
Substitute h = 0 into equation (3):
0 * a = 12
0 = 12
Since this is not possible, it must mean that there was an error. Let us try again with the calculated values.
Now we have:
12 = 25a + [k]
Since h = 0:
12 = 25a + [k]
12 = 25a + [k]
From equation (1), we have a = 12/25:
12 = 25(12/25) + [k]
12 = 12 + [k]
0 = [k]
So, Charlie's profit function p(x) is:
p(x) = (12/25)x^2
Therefore, [k] = 0.