If 400 people will attend a carnival when the admission price is $3.00, and if for every $0.10 increase in price, 10 fewer people attend, what admission price would yield the greatest revenue?

Question

I also have another questions

How many people will attend at this price?

*Also please dont use calculus*

Please help, i cant get the answer! Thank you in advance

Answer 1

First off, it's important to note that the total revenue comes from multiplying the number of people that attend the carnival by the admission price for each person. Therefore, your function that will model the revenue earned will involve the product of (people) x (admission price). The function that would model this would be f(x) = (400 - 10x) * (3 + 0.10x). In this function, (400 - 10x) represents the number of people that attend the carnival while (3 + 0.10x) represents the price of admission. Both the number of people and the admission price are dependent the variable x, and as x is incremented by 1, the number of carnival goers decreases by 10 while the admission price increases by $0.10. When you multiply (400 - 10x) * (3 + 0.10x), we get the quadratic function 1200 + 10x - x². Since the coefficient of the x² term is negative, the function represent a concave down parabola, which means that the function has a maximum value (or a peak). This max value of the function indicates the maximum revenue that can be earned given a corresponding value of x. In order to find this value of x, you must determine the vertex of this parabola using the formula -b/2a. In this case, b = 10 and a = -1, so the vertex of the function is on x=5. However, you aren't finished here, since you must still find the admission price that corresponds with this maximum value. Therefore, you must plug this value of x into your variable that represents the admission price, which was (3 + 0.10x). The final answer should come out to be $3.50