Question 1171433
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.