SOLUTION: You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by p = 40 − 4q2where q is the number of lightweight compasse

Question 1168305: You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by p = 40 − 4q2where 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 solve this problem step by step.
**Given:**
* Demand function: $p = 40 - 4q^2$ (where $p$ is the price per compass and $q$ is the quantity produced in millions)
* Cost per compass: $15
**Part (i) Write an equation giving profit as a function of the number of lightweight compasses produced.**
1. **Revenue Function (R):**
Revenue is the total money earned from selling the compasses. It is the product of the price per compass and the quantity sold (which we assume is equal to the quantity produced, $q$). Since $q$ is in millions, the revenue will be in millions of dollars.
$R(q) = p \times q = (40 - 4q^2) \times q = 40q - 4q^3$ (in millions of dollars)
2. **Cost Function (C):**
The cost of producing the compasses is the cost per compass multiplied by the quantity produced. Since $q$ is in millions and the cost per compass is in dollars, the total cost will be in millions of dollars.
$C(q) = 15 \times q = 15q$ (in millions of dollars)
3. **Profit Function (P):**
Profit is the difference between the total revenue and the total cost.
$P(q) = R(q) - C(q) = (40q - 4q^3) - 15q = -4q^3 + 25q$ (in millions of dollars)
So, the equation giving profit as a function of the number of lightweight compasses produced (in millions) is:
$\boxed{P(q) = -4q^3 + 25q}$
**Part (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 the current profit:**
Substitute $q = 2$ (million) into the profit function:
$P(2) = -4(2)^3 + 25(2) = -4(8) + 50 = -32 + 50 = 18$
This confirms that producing 2 million compasses yields a profit of $18 million.
2. **Set the profit function equal to the target profit:**
We want to find a smaller value of $q$ such that $P(q) = 18$:
$-4q^3 + 25q = 18$
3. **Rearrange the equation:**
$-4q^3 + 25q - 18 = 0$
4. **Solve the cubic equation:**
We know that $q = 2$ is one solution to this equation. This means that $(q - 2)$ is a factor of the polynomial $-4q^3 + 25q - 18$. We can use polynomial division or synthetic division to find the other factors.
Using synthetic division with the root $q = 2$:
```
2 | -4 0 25 -18
| -8 -16 18
---------------------
-4 -8 9 0
```
The resulting quadratic factor is $-4q^2 - 8q + 9$.
5. **Solve the quadratic equation:**
We need to find the roots of $-4q^2 - 8q + 9 = 0$. We can use the quadratic formula:
$q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Here, $a = -4$, $b = -8$, and $c = 9$.
$q = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(-4)(9)}}{2(-4)}$
$q = \frac{8 \pm \sqrt{64 + 144}}{-8}$
$q = \frac{8 \pm \sqrt{208}}{-8}$
$q = \frac{8 \pm \sqrt{16 \times 13}}{-8}$
$q = \frac{8 \pm 4\sqrt{13}}{-8}$
$q = -1 \pm \frac{-\sqrt{13}}{2}$
The two other possible values for $q$ are:
$q_1 = -1 - \frac{\sqrt{13}}{2} \approx -1 - \frac{3.606}{2} \approx -1 - 1.803 = -2.803$
$q_2 = -1 + \frac{\sqrt{13}}{2} \approx -1 + \frac{3.606}{2} \approx -1 + 1.803 = 0.803$
6. **Identify the smaller positive value:**
We are looking for a smaller number of lightweight compasses than the current production of 2 million. We need a positive value for $q$. The positive values we found are $q = 2$ and $q \approx 0.803$. The smaller of these is approximately $0.803$.
Since $q$ represents the number of compasses produced in millions, the smaller number of lightweight compasses the company could produce to yield the same profit is approximately $0.803$ million.
Rounding to a reasonable number of decimal places, we can say approximately 0.8 million.
Final Answer: The final answer is:
(i) $\boxed{P(q) = -4q^3 + 25q}$
(ii) $\boxed{0.803 \text{ million}}$
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