The problem gives a production function $q = K^{1/2} L^{1/3}$, where $q$ is the quantity of output, $K$ is capital, and $L$ is labor. In the short run, capital is fixed at $K = 100$. The wage rate is $w = $10$, and the rental rate on capital is $r = $20$. We want to find the short-run marginal cost (MC).

1. Problem Description

The problem gives a production function $q = K^{1/2} L^{1/3}$, where $q$ is the quantity of output, $K$ is capital, and $L$ is labor. In the short run, capital is fixed at $K = 100$. The wage rate is $w =$10$, and the rental rate on capital is$r = $20$. We want to find the short-run marginal cost (MC).

2. Solution Steps

First, substitute $K = 100$ into the production function:

$q = (100)^{1/2} L^{1/3} = 10 L^{1/3}$.

Solve for $L$ in terms of $q$:

$L^{1/3} = \frac{q}{10}$

$L = (\frac{q}{10})^3 = \frac{q^3}{1000}$.

The short-run total cost (STC) is the sum of the variable cost (labor cost) and the fixed cost (capital cost).

$STC = wL + rK$

$STC = 10L + 20(100)$

Substitute $L = \frac{q^3}{1000}$ into the STC equation:

$STC = 10(\frac{q^3}{1000}) + 2000$

$STC = \frac{q^3}{100} + 2000$

The marginal cost (MC) is the derivative of the short-run total cost (STC) with respect to quantity $q$:

$MC = \frac{d(STC)}{dq} = \frac{d}{dq}(\frac{q^3}{100} + 2000)$

$MC = \frac{3q^2}{100}$

3. Final Answer

The short-run marginal cost (MC) is $\frac{3q^2}{100}$.

The correct answer is a. $3q^2/100$.