Question 1168416
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.