Question

Type the correct answer in each blank. Use numerals instead of words. If necessary, use / for the fraction bar(s). John hosts an art workshop on the weekends. He has an average of 14 students in each session and charges a fee of $12 per session. He estimates that for every $2 increases in the fee, the average number of students reduces by 1. Complete the equation that models this scenario, where c(x) is the revenue generated and x is the number of $2 fee increases.
c(x) =_x^2 +_x+_

Answer 1

We know that John is hosting an art workshop on the weekends. He has an average of 14 students in each session and charges a fee of $12 per session. . He estimates that for every $2 increases in the fee, the average number of students reduces by 1. 

Therefore, we can conclude that his revenue is $14 * 12$ or $168$ bucks. 

The session price goes up in a pattern of 2. 

For example, from 12 the session price goes to 14, 16, 18 and so on.  

As each jumps goes by 2, the amount of students drop by 1. 

(i.e from 14 students the number decreases to 13, to 12 and so on.) 

As we can see the revenue begins at 168 and continues upward until it reaches 200 dollars and then starts to descend.

This indicates that the U-turn or vertex of the revenue function is approximately 4200, where h represents 4, and k is 200. 

To solve we can use Parabola. 

⇒  Parabola: any point is at an equivalent separation from: 

a settled point (the concentration ), and 

a settled straight line (the directrix )

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$f (x) = a(x - h)^2 + k$, where (h, k) is the vertex of the parabola.

⇒ $\left \{ {{y=192} \atop {x=2}} \right.$ 

⇒ $192 = a(2-4)^2+200$ 

⇒ $-8=a(-2)^2$ 

⇒ $-8=a*2^2$ 

⇒ $-8=a*4$ 

⇒ $-8=4a$ 

⇒ $-\frac{8}{4} = \alpha$ 

⇒ $\alpha = -2$ 

Now lets do the final step: Simplifying 

$y-2(x-4)^2+200$ 

$y=-2(x^2-8x+16)+200$ 

$y=-2x^2+16x-32+200$ 

$c(x) = -2x^2+16x+168$