Question

Could i have some quick quick help?

Answer 1

Turn over into vertex form

• y=-30t²+450t-790
• y=-30(t²-15x+79/3)

Solving

• y=-30[(t-15/2)²-359/12]

Open brackets

• y=-30(t-15/2)²+1795/2

Accurating

• y=-30(t-7.5)²+897..5

Compare to Vertex form y=a(x-h)²+k

Vertex

• (h,k)=7.5,897.5

Max profit is $897.5

• ticket price should be $7.5

Answer 2

Answer:

$7.50

Step-by-step explanation:

Completing the square formula

$\begin{aligned}y & =ax^2+bx+c\\& =a\left(x^2+\dfrac{b}{a}x\right)+c\\\\& =a\left(x^2+\dfrac{b}{a}x+\left(\dfrac{b}{2a}\right)^2\right)+c-a\left(\dfrac{b}{2a}\right)^2\\\\& =a\left(x-\left(-\dfrac{b}{2a}\right)\right)^2+c-\dfrac{b^2}{4a}\end{aligned}$

$\begin{aligned}P & =-30t^2+450t-790\\& =-30\left(t^2+\dfrac{450}{-30}t\right)-790\\\\& =-30\left(t^2+\dfrac{450}{-30}t+\left(\dfrac{450}{2(-30)}\right)^2\right)-790-(-30)\left(\dfrac{450}{2(-30)}\right)^2\\\\& =-30\left(t-\left(-\dfrac{450}{2(-30)}\right)\right)^2-790-\dfrac{450^2}{4(-30)}\\\\& =-30(t-7.5)^2+897.5\end{aligned}$

Therefore, the vertex is (7.5, 897.5)

So the ticket price that maximizes daily profit is the x-value of the vertex:  $7.50