Answer:
she should charge $7.395 per ticket in order to make the most money.
Step-by-step explanation:
From the given information:
If Rosalie charges $5 then 1175 people will attend the circus performance
If Rosalie charges $7 then 935 people will attend the circus performance
Let x be the cost and y to be the number of people that will attend the performance . Then, we will have two points which are;
(5, 1175) and (7, 935)
The slope(m) of this points = [tex]\dfrac{\Delta y}{\Delta x}[/tex]
= [tex]\dfrac{y_2-y_1}{x_2-x_1}[/tex]
[tex]=\dfrac{935-1175}{7-5}[/tex]
Slope (m) = [tex]\dfrac{-240}{2}[/tex]
Slope (m) = -120
However; we can now have the linear equation:
[tex]y-y_1 = m(x-x_1)[/tex]
[tex]y-1175= -120(x-5)[/tex]
[tex]y-1175= -120x+600[/tex]
[tex]y= -120x+600+1175[/tex]
[tex]y= -120x+1775[/tex]
The linear function is : y = -120x + 1775
Now; the total amount of money she can now earn is:
f(x) = xy
f(x) = x(-120x + 1775)
f(x) = -120x² + 1775x
The above expression is a quadratic equation; Using the quadratic formula; we have:
[tex]=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]
where; a = -120 ; b = +1775 and c = 0
[tex]=\dfrac{-(1775) \pm \sqrt{(1775)^2-4(-120)(0)}}{2(-120)}[/tex]
[tex]=\dfrac{-(1775) + \sqrt{(1775)^2-4(-120)(0)}}{2(-120)} \ \ \ \ \ OR \ \ \ \ \dfrac{-(1775) -\sqrt{(1775)^2-4(-120)(0)}}{2(-120)}[/tex]
[tex]=\dfrac{-(1775) + \sqrt{(1775)^2}}{(-240)} \ \ \ \ \ OR \ \ \ \ \dfrac{-(1775) -\sqrt{(1775)^2}}{(-240)}[/tex]
[tex]=\dfrac{-(1775) + (1775)}{(-240)} \ \ \ \ \ OR \ \ \ \ \dfrac{-(1775) - (1775)}{(-240)}[/tex]
[tex]=\dfrac{0}{(-240)} \ \ \ \ \ OR \ \ \ \ \dfrac{-3550}{(-240)}[/tex]
= 0 OR 14.79
Since; we are considering the value greater than zero
x = 14.79
maximum value of x = 14.79/2 = 7.395
Thus ; she should charge $7.395 per ticket in order to make the most money.
