Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. the cost per person is $30 minus $0.25 for every ticket sold. Gas and other miscellaneous costs are $200.
Part a) Write the revenue and cost functions for this situation
Part b)Write the profit function for this situation
Part c)find the number of tickets needed to maximize profit and give the maximum profit
Part a)
Let's define the variables for this situation:
- The number of tickets sold: x
- The cost per person: $30 - $0.25x
- The number of group members: G (20 ≤ G ≤ 70)
- The total cost: $200
Revenue function:
The revenue is derived from the number of tickets sold multiplied by the cost per person. Since the cost per person reduces by $0.25 for each ticket sold, the revenue function would be:
Revenue(x) = x * (30 - 0.25x)
Cost function:
The total cost consists of the fixed cost of $200 plus the variable cost, which is the cost per person multiplied by the number of group members:
Cost(G) = 200 + G * (30 - 0.25x)
Part b)
Profit function:
The profit is calculated by subtracting the cost function from the revenue function:
Profit(x) = Revenue(x) - Cost(G)
Part c)
To find the number of tickets needed to maximize profit, we need to determine the derivative of the profit function with respect to x and then set it equal to zero. By solving this equation, we can find the value of x that maximizes profit.
dProfit/dx = d(Revenue(x))/dx - d(Cost(G))/dx
To find the maximum profit, we set dProfit/dx = 0 and solve for x:
0 = d(Revenue(x))/dx - d(Cost(G))/dx
Once we have the optimal value for x, we can substitute it into the profit function to calculate the maximum profit.