3)
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 function of the number of bracelets he sells.
p(x)= [a](x−[h])2+[k]
Find [a]
(1 point)
Responses
−3/4
3/4
To find the value of "a" in the profit function, we need to use the given information that Charlie will have a maximum profit of $12 when he sells 5 bracelets. This forms a vertex of the parabola represented by the profit function.
At the vertex of a parabola represented by the function p(x) = ax^2 + bx + c, the x-coordinate can be found using the formula x = -b/(2a) and the corresponding y-coordinate can be found by substituting this x-value into the function.
In this case, we know that the vertex point is (5, 12). So we can set up the equations:
5 = -b/(2a)
12 = a(5)^2 + b(5) + c
Since we are looking for the value of "a", let's solve the first equation for b:
5 = -b/(2a)
10a = -b
b = -10a
Substitute this into the second equation:
12 = a(5)^2 + (-10a)(5) + c
12 = 25a - 50a + c
12 = -25a + c
Now we can use these equations to solve for the value of "a":
-25a + c = 12
c = 25a - 12
Substitute this back into the profit function equation:
p(x) = ax^2 + bx + c
p(x) = ax^2 + (-10a)x + (25a - 12)
Since Charlie's profit can be modelled as a function of the number of bracelets he sells, the profit function is:
p(x) = ax^2 - 10ax + 25a - 12
No further simplification can be done without additional information.