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
Algebra.Com
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.
RELATED QUESTIONS
A pension fund manager decides to invest a total of at most ​$40 million in U.S.... (answered by Theo)
The girls’ softball team is sponsoring a fund-raising trip to see a professional baseball (answered by ankor@dixie-net.com)
Jerry signed up for a charity race in which he had to pay an entry fee of $35. BEfore the (answered by Tiffi0116,MathTherapy)
A carpenter charges a set fee for each job in addition to his hourly charge. If he bills (answered by ptaylor)
paul works as a tutor. For each hour he works, he charges $15. He also charges an... (answered by ewatrrr)
The service committee wants to organize a fund raising dinner. The cost of renting a... (answered by ikleyn)
A ticket agency sells tickets to a professional basketball game. The agency charges... (answered by josmiceli)
A ticket agency sells tickets to a professional basketball game. The agency charges... (answered by jorel555)
A carpenter charges a set fee for each job in addition to his hourly charge. If a... (answered by ankor@dixie-net.com)