ineedhelpnow said:
How do i deal with this problem:
A movie theater has been charging 10 dollars per person and selling 400 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every 50 cents that they lower the price, the number of movie goers will increase by 50 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at 8 dollars.
First, we need to find the demand function. The demand function, in this case, is the price a company needs to charge in order to sell $x$ amount of product. In this theater example, we are given that, at 10 dollars per person, they sell around 400 tickets. So, for the current demand, we get that $p(400) = 10$. But that's not the general demand function. The theater suspects that if they decrease the price by 50 cents, the attendance will increase by 50. This is the slope, so $\frac{-0.5}{50} = -\frac{1}{100}$. Hence, the demand function will be: $p(x) = -\frac{x}{100} + b$ where $b$ is our y-intercept. To find $b$, we will use the pair we're already given, $(400, 10)$. Then, $p(400) = -\frac{400}{100} + b = 10 \iff -4 + b = 10 \iff b = 14$. So, our demand function is $p(x) = -\frac{x}{100} + 14$.
Now, we need to calculate the consumer surplus, which is basically the price people expect to pay versus what they actually pay. The integral we need to use is:
$C_s(x) = \int_0^X [p(x) - P] ~dx$
where $X$ is the current number of tickets being sold and $P$ is the current selling price. At $8$ dollars, we get that $-\frac{x}{100} + 14 = 8 \iff -\frac{x}{100} = -6 \iff x = 600$. So, plugging in what we know, we get:
$C_s(x) = \int_0^{600}\left[-\frac{x}{100} + 14 - 8\right]~dx$
$= \int_0^{600}\left[-\frac{x}{100} + 6\right] ~dx$
$= \left[-\frac{x^2}{200} + 6x\right]^{600}_{0}$
$= -1800 + 3600$
$= 1800$
So, the consumer surplus is around 1800 dollars if the price is set at 8 dollars.