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 = ![\dfrac{\Delta y}{\Delta x}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D)
= ![\dfrac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=%5Cdfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
![=\dfrac{935-1175}{7-5}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B935-1175%7D%7B7-5%7D)
Slope (m) = ![\dfrac{-240}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7B-240%7D%7B2%7D)
Slope (m) = -120
However; we can now have the linear equation:
![y-y_1 = m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%20%3D%20m%28x-x_1%29)
![y-1175= -120(x-5)](https://tex.z-dn.net/?f=y-1175%3D%20-120%28x-5%29)
![y-1175= -120x+600](https://tex.z-dn.net/?f=y-1175%3D%20-120x%2B600)
![y= -120x+600+1175](https://tex.z-dn.net/?f=y%3D%20-120x%2B600%2B1175)
![y= -120x+1775](https://tex.z-dn.net/?f=y%3D%20-120x%2B1775)
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:
![=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B-b%20%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
where; a = -120 ; b = +1775 and c = 0
![=\dfrac{-(1775) \pm \sqrt{(1775)^2-4(-120)(0)}}{2(-120)}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B-%281775%29%20%5Cpm%20%5Csqrt%7B%281775%29%5E2-4%28-120%29%280%29%7D%7D%7B2%28-120%29%7D)
![=\dfrac{-(1775) + \sqrt{(1775)^2-4(-120)(0)}}{2(-120)} \ \ \ \ \ OR \ \ \ \ \dfrac{-(1775) -\sqrt{(1775)^2-4(-120)(0)}}{2(-120)}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B-%281775%29%20%2B%20%20%5Csqrt%7B%281775%29%5E2-4%28-120%29%280%29%7D%7D%7B2%28-120%29%7D%20%20%5C%20%5C%20%5C%20%5C%20%5C%20OR%20%20%5C%20%5C%20%5C%20%5C%20%20%5Cdfrac%7B-%281775%29%20-%5Csqrt%7B%281775%29%5E2-4%28-120%29%280%29%7D%7D%7B2%28-120%29%7D)
![=\dfrac{-(1775) + \sqrt{(1775)^2}}{(-240)} \ \ \ \ \ OR \ \ \ \ \dfrac{-(1775) -\sqrt{(1775)^2}}{(-240)}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B-%281775%29%20%2B%20%20%5Csqrt%7B%281775%29%5E2%7D%7D%7B%28-240%29%7D%20%20%5C%20%5C%20%5C%20%5C%20%5C%20OR%20%20%5C%20%5C%20%5C%20%5C%20%20%5Cdfrac%7B-%281775%29%20-%5Csqrt%7B%281775%29%5E2%7D%7D%7B%28-240%29%7D)
![=\dfrac{-(1775) + (1775)}{(-240)} \ \ \ \ \ OR \ \ \ \ \dfrac{-(1775) - (1775)}{(-240)}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B-%281775%29%20%2B%20%20%281775%29%7D%7B%28-240%29%7D%20%20%5C%20%5C%20%5C%20%5C%20%5C%20OR%20%20%5C%20%5C%20%5C%20%5C%20%20%5Cdfrac%7B-%281775%29%20-%20%281775%29%7D%7B%28-240%29%7D)
![=\dfrac{0}{(-240)} \ \ \ \ \ OR \ \ \ \ \dfrac{-3550}{(-240)}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B0%7D%7B%28-240%29%7D%20%20%5C%20%5C%20%5C%20%5C%20%5C%20OR%20%20%5C%20%5C%20%5C%20%5C%20%20%5Cdfrac%7B-3550%7D%7B%28-240%29%7D)
= 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.