Planner for a festival is trying to decide what to charge for ticket prices. Based on sampling they determine that if they charge $400 for a weekend pass, 40000 people will attend. However it has been pointed out to them that for each $10 reduction in price 4000 more people will attend.
a) create a quadratic relation to model this scenario.
b) Using algebra determine the break even points and explain their meaning.
c)Using algebra determine the maximum revenue the festival can make, the # of people that will attend for maximum revenue, and the price that they will charge to earn the maximum revenue.
d) determine the initial value(y intercept) and explain its meaning.
a) To create a quadratic relation to model this scenario, let's define the variables:
- x: the amount of price reduction in dollars
- y: the number of people attending the festival
We have the following given information:
- If there is no price reduction (x = 0), 40000 people will attend (y = 40000).
- For each $10 reduction in price (x), 4000 more people will attend (y).
To create a quadratic relation, we'll start by finding the initial equation when there is no price reduction:
y = 40000
Then, since for each $10 reduction in price, 4000 more people attend, we can modify the equation as follows:
y = 40000 + 4000x
However, we are given that the original price for a weekend pass is $400, so we need to scale the equation accordingly. If we let x equal the number of $10 reductions from the original price of $400, then the equation becomes:
y = 40000 + 4000(x/10)
Simplifying further, we have:
y = 40000 + 400(x/10)
y = 40000 + 40x
Therefore, the quadratic relation to model this scenario is:
y = 40000 + 40x
b) To determine the break-even points, we need to find the x-values (price reductions) where the number of people attending is equal to the original attendance when no price reduction occurs (40000 people).
Setting y equal to 40000, we can solve for x:
40000 + 40x = 40000
40x = 0
x = 0
So the break-even point is when there is no price reduction (x = 0). This means that the original ticket price of $400 will result in the same attendance of 40000 people.
c) To determine the maximum revenue, we need to find the number of people attending and the ticket price at which the revenue is maximized. Revenue is calculated by multiplying the number of tickets sold by the price.
The revenue equation is given by:
Revenue = y * (400 - 10x)
To find the maximum revenue, we can substitute the equation for y:
Revenue = (40000 + 40x) * (400 - 10x)
We can now determine the number of people attending for maximum revenue by finding the x-value that maximizes the revenue. We can do this by finding the vertex of the quadratic equation.
To simplify the equation, let's expand it first:
Revenue = 16000000 + 40000x - 4000x^2 - 4000x
Rearranging terms, we get:
Revenue = -4000x^2 + 36000x + 16000000
To find the x-value for the maximum revenue, we can use the formula x = -b / (2a), where a = -4000 and b = 36000:
x = -36000 / (2 * -4000)
x = -36000 / -8000
x = 4.5
So, the maximum revenue occurs when x = 4.5, which represents a $45 reduction from the original price.
To find the number of people attending for maximum revenue, we can substitute x = 4.5 into the y equation:
y = 40000 + 40(4.5)
y = 40000 + 180
y = 40180
Therefore, the maximum revenue is achieved when 40180 people attend, with a ticket price reduction of $45.
d) To determine the initial value (y-intercept), we need to find the value of y when x = 0. Substituting x = 0 into the y equation, we get:
y = 40000 + 40(0)
y = 40000
The initial value (y-intercept) is 40000, which represents the number of people attending when there is no price reduction. In this scenario, it corresponds to a ticket price of $400, the original price for a weekend pass.