The problem Q-3(a) asks us to find the output level ($x$) at which a firm's profit is maximized, given the demand function $P = 500 - 0.2x$ and the cost function $C = 25x + 10000$. Additionally, we must determine the price ($P$) the firm will charge at this optimal output level.

1. Problem Description

The problem Q-3(a) asks us to find the output level ($x$) at which a firm's profit is maximized, given the demand function $P = 500 - 0.2x$ and the cost function $C = 25x + 10000$. Additionally, we must determine the price ($P$) the firm will charge at this optimal output level.

2. Solution Steps

a) To maximize profit, we first need to find the profit function. Profit ($\pi$) is defined as total revenue (TR) minus total cost (TC), where TR is price (P) times output (x).

$TR = P \cdot x = (500 - 0.2x)x = 500x - 0.2x^2$

$\pi = TR - TC = (500x - 0.2x^2) - (25x + 10000) = -0.2x^2 + 475x - 10000$

To find the output level that maximizes profit, we take the derivative of the profit function with respect to $x$ and set it equal to zero.

$\frac{d\pi}{dx} = -0.4x + 475$

Setting $\frac{d\pi}{dx} = 0$:

$-0.4x + 475 = 0$

$0.4x = 475$

$x = \frac{475}{0.4} = 1187.5$

To ensure that this is a maximum, we take the second derivative of the profit function:

$\frac{d^2\pi}{dx^2} = -0.4$

Since the second derivative is negative, we have a maximum at $x = 1187.5$.

b) Now we need to find the price at this output level. We use the demand function:

$P = 500 - 0.2x = 500 - 0.2(1187.5) = 500 - 237.5 = 262.5$

3. Final Answer

a) The output at which the profits of the firm are maximum is 1187.

5. b) The price it will charge is 262.

5.