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 a
To find the value of a in the profit function p(x) = a(x-h)^2 + k, we need to use the given information that Charlie will have a maximum profit of $12 when he sells 5 bracelets.
Since the profit function is a quadratic function and the maximum profit occurs at the vertex of the parabola, we can use the vertex form of a quadratic function:
The vertex form of a quadratic function is given by: f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.
Given that the maximum profit is $12 when Charlie sells 5 bracelets, we have:
p(5) = 12
Substitute x=5 into the profit function:
p(5) = a(5-h)^2 + k = 12
Since we are only trying to find the value of a, k is not needed for this calculation.
Solving for a, we get:
a(5-h)^2 = 12
a(25-10h+h^2) = 12
25a - 10ah + ah^2 = 12
Simplify this equation to get:
ah^2 - 10ah + 25a = 12
Since we are given that the maximum profit occurs at the vertex, we know that the vertex of the parabola occurs at the point (5, 12) which can be used to find h.
Substitute x=5 into the profit function:
p(5) = a(5-h)^2 + k = 12
a(5-h)^2 + k = 12
Given that Charlie's maximum profit is $12 when he sells 5 bracelets, the vertex of the parabola is (5, 12).
So, h = 5.
Substitute h=5 into the equation to solve for a:
5a^2 - 50a + 25a = 12
5a^2 - 25a = 12
a(5a - 25) = 12
5a^2 - 25a - 12 = 0
This is a quadratic equation that can be solved using the quadratic formula or any other method you are comfortable with.
After solving for a, you will find the value of a in the profit function p(x) = a(x-h)^2 + k.
FIND A OR THE VALUE OF A
To find the value of "a" in the profit function p(x) = a(x-h)^2 + k, we will first substitute the known values into the equation and then solve for "a".
Given:
- The profit function is p(x) = a(x-h)^2 + k
- Charlie will have a maximum profit of $12 when he sells 5 bracelets
- The vertex of the parabola is at the point (5, 12) which means h = 5
- Therefore, p(5) = a(5-5)^2 + k = 12
- Simplifying, we get: 0 + k = 12, which means k = 12
So, we have the updated profit function: p(x) = a(x-5)^2 + 12
Now, we need more information to find the value of "a". The given data doesn't directly provide enough information to find "a". If there's further information or if you have a specific relationship or additional point that can help determine the value of "a", please provide it.