The image presents a question about a company's cost function and demand function. Let's solve it step by step:

Given:

• Total Cost (TC) Function: $TC = 50 + 3Q$
• Demand Function: $P = 100 - 2Q$

Where $Q$ is the quantity produced and sold, and $P$ is the price.

a. Total Revenue (TR) Function:

Total revenue is calculated by multiplying the price per unit (P) by the quantity sold (Q).

$TR = P \times Q$

Substitute the price function $P = 100 - 2Q$ into the revenue function:

$TR = (100 - 2Q) \times Q$ $TR = 100Q - 2Q^2$

So, the total revenue function is:

$TR(Q) = 100Q - 2Q^2$

b. Profit Function:

Profit ($\pi$) is calculated as total revenue minus total cost:

$\pi(Q) = TR(Q) - TC(Q)$

Substitute the total revenue and total cost functions:

$\pi(Q) = (100Q - 2Q^2) - (50 + 3Q)$ $\pi(Q) = 100Q - 2Q^2 - 50 - 3Q$ $\pi(Q) = 97Q - 2Q^2 - 50$

Thus, the profit function is:

$\pi(Q) = 97Q - 2Q^2 - 50$

c. Breakeven Point:

The breakeven point occurs when profit is zero, i.e., when total revenue equals total cost:

$TR(Q) = TC(Q)$

From the total revenue function and total cost function:

$100Q - 2Q^2 = 50 + 3Q$ Rearrange the equation:

$-2Q^2 + 100Q - 3Q - 50 = 0$ $-2Q^2 + 97Q - 50 = 0$

This is a quadratic equation in the form $ax^2 + bx + c = 0$. We can solve it using the quadratic formula:

$Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = -2$, $b = 97$, and $c = -50$:

$Q = \frac{-97 \pm \sqrt{97^2 - 4(-2)(-50)}}{2(-2)}$ $Q = \frac{-97 \pm \sqrt{9409 - 400}}{-4}$ $Q = \frac{-97 \pm \sqrt{9009}}{-4}$ $Q = \frac{-97 \pm 94.95}{-4}$

Now calculate both roots:

1. $Q_1 = \frac{-97 + 94.95}{-4} = \frac{-2.05}{-4} = 0.5125$
2. $Q_2 = \frac{-97 - 94.95}{-4} = \frac{-191.95}{-4} = 47.9875$

Thus, the breakeven points occur when $Q = 0.5125$ and $Q = 47.9875$.

Summary of Answers:

• a. Total Revenue Function: $TR(Q) = 100Q - 2Q^2$
• b. Profit Function: $\pi(Q) = 97Q - 2Q^2 - 50$
• c. Breakeven Points: $Q = 0.51$ and $Q = 47.99$

Would you like further details or clarifications on any part of the solution?

Here are 5 questions you could explore further:

1. What happens to profit if the production quantity increases beyond the breakeven points?
2. How would a change in fixed costs affect the breakeven points?
3. What is the marginal revenue and marginal cost based on these functions?
4. How can we interpret the breakeven quantities in a real-world context?
5. What impact would price elasticity of demand have on this company's revenue function?

Tip: In optimization problems, finding the breakeven points is key to understanding the minimum and maximum levels of production for profitability.