Question 1176522: 1. You sell peanuts at UNK athletic events to make some extra money. When peanuts are sold for $1.00 per bag, approximately 600 bags are sold at each event. You tried to raise the price to $1.25 and found that the quantity demanded dropped to 560 bags. The startup cost was $500 and the cost to you per bag of peanuts is $0.50.
A. Assume that the demand function is linear and write a function that models the profit from selling x bags of peanuts at an event.
B. What number of bags and what price per bag will get you a maximum profit?
C. Does your answer to part B agree with what you "thought" it would be? Why or why not?
Even telling me what kind of question this is so I can find other examples would be helpful, thanks!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**A. Model the Profit Function**
1. **Find the Demand Function (p(x))**
* We have two points: (600, 1.00) and (560, 1.25)
* Find the slope (m):
* m = (1.25 - 1.00) / (560 - 600) = 0.25 / -40 = -0.00625
* Use point-slope form (y - y1 = m(x - x1)):
* p - 1.00 = -0.00625(x - 600)
* p = -0.00625x + 3.75 + 1.00
* p(x) = -0.00625x + 4.75
2. **Find the Revenue Function (R(x))**
* Revenue = price * quantity
* R(x) = x * p(x)
* R(x) = x(-0.00625x + 4.75)
* R(x) = -0.00625x² + 4.75x
3. **Find the Cost Function (C(x))**
* Cost = startup cost + cost per bag * quantity
* C(x) = 500 + 0.50x
4. **Find the Profit Function (P(x))**
* Profit = Revenue - Cost
* P(x) = R(x) - C(x)
* P(x) = (-0.00625x² + 4.75x) - (500 + 0.50x)
* P(x) = -0.00625x² + 4.25x - 500
**B. Maximize Profit**
1. **Find the Vertex of the Profit Function**
* The profit function is a quadratic, so its maximum occurs at the vertex.
* The x-coordinate of the vertex is given by x = -b / 2a, where a = -0.00625 and b = 4.25.
* x = -4.25 / (2 * -0.00625)
* x = -4.25 / -0.0125
* x = 340
2. **Find the Price per Bag**
* p(x) = -0.00625x + 4.75
* p(340) = -0.00625(340) + 4.75
* p(340) = -2.125 + 4.75
* p(340) = 2.625
3. **Find the Maximum Profit**
* P(x) = -0.00625x² + 4.25x - 500
* P(340) = -0.00625(340)² + 4.25(340) - 500
* P(340) = -0.00625(115600) + 1445 - 500
* P(340) = -722.5 + 1445 - 500
* P(340) = 222.5
**Answers for B:**
* Number of bags: 340
* Price per bag: $2.625 (or $2.63)
**C. Agreement with Intuition**
* **Initial Intuition:** You might have thought that raising the price would always decrease the number of bags sold and potentially lower the profit.
* **Actual Result:** The analysis shows that there's an optimal price point that maximizes profit. In this case, raising the price significantly above the initial $1.00 leads to a lower quantity sold but a higher profit.
* **Explanation:** This is because the higher price per bag more than compensates for the reduced quantity sold, up to a certain point. The profit function is a parabola, and the vertex represents the optimal balance between price and quantity.
**Why the "thought" might not be what the answer is:**
* **Linear Demand Assumption:** The linear demand function is a simplification. Real-world demand might not be perfectly linear.
* **Cost Structure:** The constant cost per bag and fixed startup cost simplify the model. Real costs might be more complex.
* **Consumer Behavior:** The model assumes rational consumer behavior. In reality, factors like brand loyalty, perceived value, and competitor pricing can influence demand.
In conclusion, the mathematical analysis provides a more precise and optimal solution than relying on initial intuition alone.
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