Question 1186827: Good Day,
I didn't get any response to this question.....The demand function for a tablet is given by the model p=200-16x^2, where p is measured in dollars per tablet and x is measured in millions of tablets. If it costs $50 to produce each tablet and a profit of $125 million was derived when 2.5 million tablets were produced. Derive the number of tablets that the company could sell to make the same amount of profit?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**1. Define the variables:**
* p = price per tablet (dollars)
* x = quantity of tablets (millions)
**2. Given information:**
* Demand function: p = 200 - 16x²
* Cost per tablet: $50
* Profit: $125 million when x = 2.5 million
**3. Find the profit function:**
* Revenue = p * x = (200 - 16x²) * x = 200x - 16x³
* Cost = 50 * x
* Profit = Revenue - Cost = (200x - 16x³) - 50x = 150x - 16x³
**4. Set up the equation to solve for x:**
We know the profit was $125 million when x = 2.5 million. We want to find the other value of x that gives the same profit.
125 = 150x - 16x³
**5. Solve for x:**
16x³ - 150x + 125 = 0
This is a cubic equation. We already know one root: x = 2.5. We can use this information to factor the equation or use numerical methods (like a calculator or software) to find the other root(s).
*One way to proceed is to use polynomial division to divide the cubic by (x - 2.5). This will leave you with a quadratic equation which you can then solve using the quadratic formula.*
Solving the cubic equation (using a calculator or software is the most practical method), we find three roots: x ≈ -3.06, x = 2.5, and x ≈ 3.56.
**6. Interpret the results:**
* x ≈ -3.06: This solution doesn't make sense in our context, as the quantity of tablets cannot be negative.
* x = 2.5: This is the given quantity, which we already know yields a profit of $125 million.
* x ≈ 3.56: This is the other solution we're looking for.
**Answer:**
The company could sell approximately 3.56 million tablets to make the same amount of profit ($125 million).
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