qns

Given Data:

• Demand function: $Q = 45 - 3P$, where $Q$ is quantity and $P$ is price.
• Fixed cost (FC): $Rs. 60$
• Variable cost (VC per unit): $Rs. 5$

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(a) Total Cost, Average Cost, and Marginal Cost

1. Total Cost (TC):

   $$TC = FC + VC$$

   Substituting the values:

   $$TC = 60 + 5Q$$

2. Average Cost (AC):

   $$AC = \frac{TC}{Q}$$

   Substituting $TC$:

   $$AC = \frac{60 + 5Q}{Q} = \frac{60}{Q} + 5$$

3. Marginal Cost (MC): Marginal cost is the derivative of $TC$ with respect to $Q$:

   $$MC = \frac{d(TC)}{dQ}$$

   Since $TC = 60 + 5Q$:

   $$MC = 5$$

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(b) Total Revenue, Average Revenue, and Marginal Revenue

1. Price function (from demand equation):

   $$P = 15 - \frac{Q}{3}$$

2. Total Revenue (TR):

   $$TR = P \cdot Q$$

   Substituting $P$:

   $$TR = \left(15 - \frac{Q}{3}\right) Q = 15Q - \frac{Q^2}{3}$$

3. Average Revenue (AR):

   $$AR = \frac{TR}{Q}$$

   Substituting $TR$:

   $$AR = \frac{15Q - \frac{Q^2}{3}}{Q} = 15 - \frac{Q}{3}$$

4. Marginal Revenue (MR): Marginal revenue is the derivative of $TR$ with respect to $Q$:

   $$MR = \frac{d(TR)}{dQ}$$

   Since $TR = 15Q - \frac{Q^2}{3}$:

   $$MR = 15 - \frac{2Q}{3}$$

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(c) Profit Function and Marginal Profit

1. Profit Function ($\pi$): Profit is $TR - TC$:

   $$\pi = TR - TC$$

   Substituting $TR = 15Q - \frac{Q^2}{3}$ and $TC = 60 + 5Q$:

   $$\pi = \left(15Q - \frac{Q^2}{3}\right) - (60 + 5Q)$$

   Simplifying:

   $$\pi = 15Q - \frac{Q^2}{3} - 60 - 5Q = 10Q - \frac{Q^2}{3} - 60$$

2. Marginal Profit ($\frac{d\pi}{dQ}$):

   $$\frac{d\pi}{dQ} = \frac{d}{dQ}\left(10Q - \frac{Q^2}{3} - 60\right)$$

   Differentiating:

   $$\frac{d\pi}{dQ} = 10 - \frac{2Q}{3}$$

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(d) Break-even Point

At the break-even point, $TR = TC$. Setting $TR = 15Q - \frac{Q^2}{3}$ and $TC = 60 + 5Q$:

$$15Q - \frac{Q^2}{3} = 60 + 5Q$$

Simplify:

$$15Q - 5Q - \frac{Q^2}{3} = 60$$ $$10Q - \frac{Q^2}{3} = 60$$

Multiply through by 3 to eliminate the fraction:

$$30Q - Q^2 = 180$$

Rearrange into standard quadratic form:

$$Q^2 - 30Q + 180 = 0$$

Solve using the quadratic formula:

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

Here, $a = 1$, $b = -30$, $c = 180$:

$$Q = \frac{-(-30) \pm \sqrt{(-30)^2 - 4(1)(180)}}{2(1)}$$ $$Q = \frac{30 \pm \sqrt{900 - 720}}{2}$$ $$Q = \frac{30 \pm \sqrt{180}}{2}$$ $$Q = \frac{30 \pm 13.42}{2}$$ $$Q = 21.71 \, \text{or} \, Q = 8.29$$

The break-even quantities are approximately $Q = 21.71$ and $Q = 8.29$.

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(e) Graph Total Costs and Total Revenue

For the graph:

1. Plot $TC = 60 + 5Q$, which is a straight line.
2. Plot $TR = 15Q - \frac{Q^2}{3}$, which is a downward-opening parabola.

The intersection points of the two curves represent the break-even points.