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.

Question

p(x) = [a] (x-[h])^2 + [k]

Find [k]

Answer 1

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.