A river boat company offers a 4th of july trip to an organization with understanding that there will be at least 400 passengers. each ticket is $12, and company agrees to refund $.20 to every passenger for each 10 passengers in excess of 400. find the number of passengers that makes the total revenue a maximum.
revenue = price * tickets
If 400+x tickets are sold,
r = (12 - .2x/10)(400+x)
r reaches max at x=100, or 500 tickets sold
A piece of wire of length 246cm is cut into 12 parts whose length are in an A.P. Given that the sum of four shortest parts is 34cm, find the length of the longest part.
To find the number of passengers that maximizes the total revenue, we need to determine the revenue function and then find its maximum point.
Let's break down the problem step-by-step:
Step 1: Express the revenue function in terms of the number of passengers.
The total revenue includes the base revenue from 400 passengers and the refund amount for every additional 10 passengers beyond 400.
Base revenue from 400 passengers = 400 * $12
Refund amount per 10 extra passengers = $0.20 * (Number of extra passengers / 10)
So, the revenue function in terms of the number of passengers (x) is:
Revenue(x) = (400 * 12) + ($0.20 * (x - 400) / 10)
Step 2: Simplify the revenue function.
Revenue(x) = 4800 + ($0.02 * (x - 400))
Revenue(x) = 4800 + $0.02x - $0.02 * 400
Revenue(x) = 4800 + $0.02x - $8
Revenue(x) = $0.02x + 4792
Step 3: Find the derivative of the revenue function.
Taking the derivative of the revenue function will give us the rate of change of revenue with respect to the number of passengers.
dRevenue/dx = $0.02
Step 4: Set the derivative equal to zero and solve for x.
To find the maximum point of the revenue function, we need to find the x-value where the derivative is equal to zero.
0.02 = 0
Since 0.02 is a non-zero constant, it does not equal zero. Therefore, the derivative does not have a maximum or minimum. Instead, it suggests that the revenue will increase indefinitely as the number of passengers increases.
Step 5: Analyze the problem further.
Given the formulation of the problem, it seems that there is no fixed maximum number of passengers that would maximize the revenue. However, there may be practical constraints or limitations imposed on the number of passengers that the river boat company can accommodate, or the organization's capacity for participants. In reality, businesses have limitations, such as seating capacity, safety regulations, or the organization's resources.
To find the number of passengers that maximizes the revenue within practical constraints, you need to consider the specific circumstances and limitations of the river boat company.