A movie theater has been charging 10.00 dollar per person and selling about 500 tickets on a typical weeknight. After surveying their customers, the theater management estimates that for every 50 cents that they lower the price, the number of moviegoers will increase by 50 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at $8.00.
Applications of Integration
Okay, so the question says that currently, a movie theatre charges $10 for tickets and they get 500 customers. However, they estimate that for every 50 cent decrease in ticket price, they could get 50 more customers. And then we want the demand function and the consumer surplus at $8 ticket price. So first for the demand function now first, for the slope of the demand function, we see that every 50 cent decrease gets 50 more customers. So for the slope of the line, we know that it is rise overrun. In this case, a 50 cent decrease is a minus 0.5 rise and then a run of 50 which is also minus one over 100. And then currently the 0.10 comma 500 is on the line. So using point slope form, we can say that p minus 10 is equal to this news slope that we found my ass one over 100 times X minus 500 and then to get the final function, isolate P and we get the minus 100 X minus one over 100 x minus one over 100 times minus 500 of us plus five and then adding The 10 gives us plus 15. So that is our demand function now for the consumer surplus. If we draw a graph, we have P. We see that from the form of P, when there are no there's new quantity, the price would be 15. It's going to go down one over 100 from there. And so now if we put in eight, we want the value on the X axis. Um, to do that, we simply take eight and said it equal to minus one over 100 x. Most 15. It's attracting 15. We get minus seven equals minus one over 100 X and then multiplying by negative 100. On both sides, we get X is equal 2 700 So now we know that point and we want our consumer surplus, which is this region up here. And since we found that our demand function is linear, we can just find the area of that region with the formula for the area of a triangle. And so with it we can find the consumer surplus. Busy people toe 1/2 the base, which is 15 of minus eight times the height, which we just found was 700 and so half a 700 is 3 50 So we get seven times 3 50 which his 2400 and $50 and that right there is the consumer surplus.