Answer :
Answer:
- Using model A, the price should be $ 5 to maximize monthly revenue because the x-intercepts occur at x= 8 and x = -2.
- Using model B, the price should be $ 6 to maximimze monthly revenue because the x-intercepts occur at x = 10 and x = -2.
Explanation:
The maximization of the monthly revenue is given by the maximum value of the function, f(x) for the model A and the curve for the model B.
1. Model A.
The function [tex]f(x)=-12.5x^2+75x+200[/tex] is a quadratic function, a parabola, whose vertex is the maximum value (since the coefficient of the quadratic term is negative, - 2).
Then, by finding the x-coordinate ot the vertex you will have the value of x (he number of $1-price increases form the original $ per bottle price) at which such maximum is reached.
To find the vertex you can transform the given function into its vertex form, which is done by completing squares:
[tex]f(x)=-12.5x^2+75x+200[/tex]
[tex]f(x)=-12.5(x^2-6x-16)[/tex]
[tex]f(x)/(-12.5)=x^2-6x-16[/tex]
[tex]f(x)/(-12.5)+16=x^2-6x[/tex]
[tex]f(x)/(-12.5)+16+9=x^2-6x+9[/tex]
[tex]f(x)/(-12.5)+25=(x-3)^2[/tex]
[tex]f(x)/(-12.5)=(x-3)^2-25[/tex]
[tex]f(x)=-12.5[(x-3)^2-25][/tex]
[tex]f(x)=12.5[(x-3)^2+312.5[/tex]
Compare with the general equation of the vertex form for a parabola:
[tex]f(x)=a(x-h)^2+k\\ \\ Where,\\ \\ h\text{ }and\text{ }k\text{ are the coordinates of the vertex}[/tex]
Hence, the vertex, the maximum, is (3, 312.5), meaning that an increase of 3 times $1-price from the original $2 per bottle price maximizes the revenue. Thus, the price is $3 + $2 = $5
The y-intercept, is always at x = 0, and it is:
[tex]f(x)=-12.5[(0-3)^2+312.5=-12.5(9)+312.5=200[/tex]
The x-intercepts are the solutions of the equation f(x) = 0.
You can find them easily by factoring the original equation:
[tex]f(x)=-12.5x^2+75x+200\\ \\ f(x)=-12.5(x^2-6x-16)=-12.5(x-8)(x+2)[/tex]
[tex]-12.5(x-8)(x+2)=0\\\\(x-8)(x+2)=0\\\\x=8\text{ }and\text{ } x=-2[/tex]
Thus, the x-intercepts occur at x = 8 and x = -2. Note that the vertex is at the symmetry axis, which is just in the middle between the two x-incercepsts:
-
- x = (8-2)/2 = 6/2 = 3, such as found above.
2. Model B.
Again, the vertex is at the symmetry axis, which is just in the middle between the two x-incercepsts.
In the graph you can see that the symmetry axis is the line x = 4:
- Midpoint between the two x-intercepts: x = (-2 + 10)/2 = 8/2 = 4.
Remenber, 4 is the number of $1 increases over the $ 2 per bottle price. Hence, for this model the price will be $4 + $2 = $6.
The y-intercept, the value of y when x = 0, is also read from the graph and it is between $ 180 and $ 210.