Question

Each year, a local schools Rock the Vote committee organizes a public rally. Based on previous years, the organizers decided that the Income from ticket sales, l(t) is related to ticket price t by the equation I(t) = 400t - 40+. a. What ticket price(s) would generate the greatest income? What is the greatest income possible? Explain how you obtained the value you got. Ticket price(s) Income b. At what ticket price(s) would there be no income from the ticket sales. Explain how you obtained the answer.

Answer 1

The given equation is

$l(t)=400t-40t^2$

The greatest income refers to the vertex of the function V(h,k), where

$h=-\frac{b}{2a}$

a = -40 and b = 400.

$h=-\frac{400}{2(-40)}=5$

Then, we find k evaluating the function

$\begin{gathered} k=400(5)-40(5)^2=2000-40(25)=2000-1000 \\ k=1000 \end{gathered}$

Hence, the greatest income is $1000. The number of tickets is 5.

The ticket price with no income refers to the x-intercept

$\begin{gathered} 400t-40t^2=0 \\ 40t(10-t)=0 \\ t=0 \\ \\ t=10 \end{gathered}$

When the tickets have a price of $10, there's no income. We obtained the answer by solving the function where l(t) = 0.