SOLUTION: Elmer is organizing a fund raising basketball tournament in their barangay. He plans to charge PHP20 entry fee for each 80 players. He recently decided to raise the entry fee by PH
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Question 1171433: Elmer is organizing a fund raising basketball tournament in their barangay. He plans to charge PHP20 entry fee for each 80 players. He recently decided to raise the entry fee by PHP5 and 5 fewer players entered with the increase. How much would Elmer charge in order to maximize the income? Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website! Let's solve this optimization problem step-by-step.
**1. Define Variables**
* Let 'x' be the number of PHP5 increases in the entry fee.
* Entry fee = 20 + 5x
* Number of players = 80 - 5x
* Income (I) = (Entry fee) * (Number of players)
**2. Formulate the Income Function**
* I(x) = (20 + 5x)(80 - 5x)
* I(x) = 1600 - 100x + 400x - 25x^2
* I(x) = -25x^2 + 300x + 1600
**3. Find the Vertex of the Quadratic Function**
The income function is a quadratic function, and its graph is a parabola that opens downward. The maximum income occurs at the vertex of the parabola.
The x-coordinate of the vertex is given by:
* x = -b / (2a)
Where:
* a = -25
* b = 300
Plugging in the values:
* x = -300 / (2 * -25)
* x = -300 / -50
* x = 6
**4. Calculate the Entry Fee**
Now, substitute the value of x back into the entry fee equation:
* Entry fee = 20 + 5x
* Entry fee = 20 + 5(6)
* Entry fee = 20 + 30
* Entry fee = 50
**5. Calculate the Number of Players**
Substitute the value of x back into the number of players equation:
* Number of players = 80 - 5x
* Number of players = 80 - 5(6)
* Number of players = 80 - 30
* Number of players = 50
**6. Calculate the Maximum Income**
* Maximum income = (Entry fee) * (Number of players)
* Maximum income = 50 * 50
* Maximum income = 2500
**Conclusion**
Elmer should charge PHP50 in order to maximize the income.
