4)
A theatre owner is trying to decide on a ticket price for his next play. The higher the price, the fewer the number of people who will buy tickets. He creates a quadratic function f(x)
to estimate his total revenue, in dollars, if he charges x
dollars per ticket. Then, he creates a table of values.
x
f(x)
0 0
10 1400
20 2000
30 1800
40 800
To create a quadratic function using the table of values provided, we need to find the equation that fits the data points.
Since we know that the function is quadratic, we can assume it has the form f(x) = ax^2 + bx + c. We can plug in the values from the table to create a system of equations:
f(0) = a(0)^2 + b(0) + c = 0
f(10) = a(10)^2 + b(10) + c = 1400
f(20) = a(20)^2 + b(20) + c = 2000
f(30) = a(30)^2 + b(30) + c = 1800
f(40) = a(40)^2 + b(40) + c = 800
From the first equation, we know that c = 0. We can substitute this into the other equations:
a(10)^2 + b(10) = 1400
a(20)^2 + b(20) = 2000
a(30)^2 + b(30) = 1800
a(40)^2 + b(40) = 800
Solving this system of equations, we get a = -2, b = 120, and c = 0. Therefore, the quadratic function that estimates the total revenue is:
f(x) = -2x^2 + 120x
Thus, the theatre owner should charge $120 per ticket to maximize revenue.