A local travel agent is organizing a charter ride to a well known resort. The agent has quoted
a price of $30 per person if 100 or fewer sign up for the ride. For every person over the 100,
the price of the ticket for all will decrease by $2.50. For instance, if 101 people sign up, each
will pay $27.50. Let x be the number of people above 100.
a) Determine the function which states price per person p = f(x), which is a function of x.
b) Formulate the revenue function R = h(x), which states the total ticket revenue R as a
function of x.
c) Plot the graph of the revenue function.
d) What value of x results in the maximum value of R?
e) What is the maximum value of R?
f) What price per ticket results in maximum R?
a) To determine the function which states the price per person p = f(x), we first need to understand the relationship between the number of people above 100 (x) and the price per person (p).
The given information tells us that for every person over 100, the price per ticket decreases by $2.50. So, initially, the price per person is $30. For each additional person above 100, the price per person decreases by $2.50.
Let's break down the relationship:
- For 100 or fewer people (x ≤ 0), the price per person is $30 (since there are no additional people).
- For each additional person (x > 0), the price per person decreases by $2.50.
Based on this, we can establish the function:
p = f(x) = $30 - $2.50x
b) To formulate the revenue function R = h(x), which states the total ticket revenue R as a function of x, we need to understand how revenue is calculated.
Revenue is calculated by multiplying the price per person (p) by the total number of people (100 + x).
The total number of people is always 100 (for the first 100 people) plus the number of additional people (x).
Therefore:
R = h(x) = p * (100 + x)
R = h(x) = ($30 - $2.50x) * (100 + x)
c) To plot the graph of the revenue function, we can use software or tools like graphing calculators, spreadsheets, or online graphing platforms. The graph will show the relationship between the number of additional people (x) and the total ticket revenue (R).
d) To find the value of x that results in the maximum value of R, we need to determine the vertex or maximum point on the graph of the revenue function. This represents the point of maximum revenue.
To find the vertex, we can use calculus or alternative methods like completing the square or graph-reading techniques. By finding the x-value of the vertex, we can determine the value of x that results in the maximum revenue.
e) Once we determine the x-value that results in the maximum revenue, we can substitute it back into the revenue function (R = h(x)) to find the maximum value of R.
f) To find the price per ticket that results in the maximum revenue, we can substitute the x-value found in part (d) back into the price per person function (p = f(x)). This will give us the price per ticket at the maximum revenue point.