SOLUTION: You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by p = 40 − 4q^2 where q is the number of lightweight compas

Question 1168416: You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by p = 40 − 4q^2 where q
is the number of lightweight compasses produced in millions. It costs the company $15 to make a lightweight compass.
(i) Write an equation giving profit as a function of the number of lightweight compasses produced.
(ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's break down this problem step by step.
**Given:**
* Demand function: p = 40 - 4q² (p is the price, q is the quantity in millions)
* Cost per compass: $15
**(i) Write an equation giving profit as a function of the number of lightweight compasses produced.**
**1. Revenue Function**
* Revenue (R) = price (p) * quantity (q)
* R = (40 - 4q²) * q
* R = 40q - 4q³
Since q is in millions, R is in millions of dollars.
**2. Cost Function**
* Cost (C) = cost per unit * quantity
* C = 15q (in millions of dollars)
**3. Profit Function**
* Profit (P) = Revenue (R) - Cost (C)
* P = (40q - 4q³) - 15q
* P = -4q³ + 25q
Therefore, the profit function is P(q) = -4q³ + 25q, where P is in millions of dollars and q is in millions of units.
**(ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?**
**1. Verify Profit at q = 2**
* P(2) = -4(2)³ + 25(2)
* P(2) = -4(8) + 50
* P(2) = -32 + 50
* P(2) = 18
This confirms the current profit of $18 million.
**2. Set Profit Function Equal to 18**
* We need to find another q value that yields a profit of 18.
* -4q³ + 25q = 18
* -4q³ + 25q - 18 = 0
**3. Solve the Cubic Equation**
We know that q = 2 is a solution, so we can factor out (q - 2).
* (-4q³ + 25q - 18) / (q - 2) = -4q² - 8q + 9
So, we have:
* (q - 2)(-4q² - 8q + 9) = 0
We need to solve the quadratic equation:
* -4q² - 8q + 9 = 0
Using the quadratic formula:
* q = [-b ± √(b² - 4ac)] / 2a
* q = [8 ± √((-8)² - 4(-4)(9))] / (2(-4))
* q = [8 ± √(64 + 144)] / -8
* q = [8 ± √208] / -8
* q = [8 ± 4√13] / -8
* q = -1 ± (-√13 / 2)
* q₁ = -1 - √13 / 2 ≈ -2.80
* q₂ = -1 + √13 / 2 ≈ 0.80
Since quantity must be positive, we take q₂ ≈ 0.80.
**4. Final Answer**
* The smaller number of lightweight compasses the company could produce to yield the same profit is approximately 0.8 million (800,000 units).
**Therefore:**
* (i) Profit function: P(q) = -4q³ + 25q
* (ii) The company could produce approximately 0.8 million lightweight compasses to yield the same profit.

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