A certain travel agency offered a tourthat will cost each person P 1500.00 if not more than 150 persons will join,however the cost per person will be reduced by P 5.00 per person in excessof 150. what will be the maximum profit?
I also find it strange that this same question appeared in that identical form back in 2016 including the same spelling and punctuation error.
https://www.jiskha.com/questions/1412396/A-certain-travel-agency-offered-a-tourthat-will-cost-each-person-P-1500-00-if-not
Well, if not more than 150 people join the tour, the travel agency will earn P 1500.00 per person. Let's calculate the maximum profit:
Maximum profit = P 1500.00 × 150 people
Maximum profit = P 225,000.00
However, if more than 150 people join, the cost per person will be reduced by P 5.00 for each additional person. Let's consider the worst-case scenario where 150 people join exactly:
Maximum profit = (P 1500.00 - P 5.00) × 150 people
Maximum profit = P 1495.00 × 150 people
Maximum profit = P 224,250.00
So, the maximum profit will be P 224,250.00. But remember, this calculation assumes that exactly 150 people will join the tour.
To determine the maximum profit, we need to find the number of people that will maximize the profit.
Let's start by considering the first scenario where the cost per person is P 1500.00 if not more than 150 persons join.
Let x represent the number of people that will join. If x <= 150, then the cost per person is P 1500.00. The total cost for x people is given by:
Cost(x) = 1500x
Next, let's consider the second scenario where the cost per person is reduced by P 5.00 per person in excess of 150.
If x > 150, the cost per person is reduced by P 5.00 for each person beyond 150. This means that the cost per person is P 1500.00 - P 5.00(x - 150). The total cost for x people is given by:
Cost(x) = (1500 - 5(x - 150))x
To find the maximum profit, we need to determine the number of people that will maximize the profit. The profit is calculated as the revenue minus the cost.
Revenue(x) = 1500x
Profit(x) = Revenue(x) - Cost(x)
To find the maximum profit, we need to find the value of x that maximizes the profit function.
To do this, we can take the derivative of the profit function with respect to x and set it equal to zero. Then solve for x.
Profit'(x) = d(Revenue(x))/dx - d(Cost(x))/dx = 0
Let's differentiate the revenue and cost functions first.
d(Revenue(x))/dx = 1500
d(Cost(x))/dx = 1500 - 5(2x - 300)
Now, let's set the derivative of the profit function equal to zero and solve for x.
Profit'(x) = 1500 - 1500 + 10x - 1500 = 0
10x - 1500 = 0
10x = 1500
x = 1500 / 10
x = 150
Therefore, the maximum profit will be achieved when 150 people join the tour.
To determine the maximum profit, we need to find the optimal number of persons who will join the tour.
First, let's calculate the maximum number of persons if the cost per person is still at P 1500.00.
Max number of persons = Total budget / Cost per person
Max number of persons = P 1500.00 / P 1500.00 = 1 person
Since the maximum number of persons cannot be less than 1, we move to the next condition.
Next, let's calculate the maximum number of persons if the cost per person is reduced by P 5.00 per person in excess of 150.
Max number of persons = 150 + (Total budget - (150 * Cost per person after discount)) / Cost per person after discount
Given:
Cost per person = P 1500.00
Cost reduction per person = P 5.00
Max number of persons = 150 + (Total budget - (150 * (Cost per person - Cost reduction per person))) / (Cost per person - Cost reduction per person)
Max number of persons = 150 + (Total budget - (150 * (1500 - 5))) / (1500 - 5)
Max number of persons = 150 + (Total budget - 150 * 1495) / 1495
To maximize the profit, we need to find the number of persons that results in the highest profit.
Profit = Total budget - (Number of persons * Cost per person)
Profit = Total budget - (Number of persons * (Cost per person - Cost reduction per person))
Profit = Total budget - (Number of persons * (1500 - 5))
We can now substitute the expression for the maximum number of persons:
Profit = Total budget - ((150 + (Total budget - 150 * 1495) / 1495) * (1500 - 5))
To find the maximum profit, we need to determine the total budget. If the total budget is not provided, we cannot calculate the exact value.
Number of people in excess of 150 ---- x
number of people on tour = x + 150
total cost = 150(1500) + x(1500-5x)
= 225000 + 1500x - 5x^2
find the vertex of this downwards opening parabola using the method you learned. The y-coordinate will be you maximum cost, (not profit as you stated)
This question makes little sense to me. If the current cost per person is 1500 each for the first 150 people, and it reduces by 5 per person for over the 150 mark, wouldn't the cost per person go down to zero if 450 people joined the tour??
that is, number of people over 150 = 300
reduction in cost per person = 300(5) = 1500
so cost = 1500-1500 = 0
check the wording of the question.