The problem asks us to minimize the cost of producing 40 units of output given the production function $q = 8L + 10K$, the wage rate $w = \$3$, and the rental rate $v = \$4$. We need to find the optimal amounts of labor (L) and capital (K) to use.

1. Problem Description

The problem asks us to minimize the cost of producing 40 units of output given the production function $q = 8L + 10K$, the wage rate w = \3$, and the rental rate$v = \$4$. We need to find the optimal amounts of labor (L) and capital (K) to use.

2. Solution Steps

Since the production function is linear, we have a linear programming problem. The goal is to minimize the cost $C = wL + vK = 3L + 4K$ subject to the constraint that $8L + 10K = 40$.

Since this is a linear production function, we can solve for the corner solutions.

First, set $L = 0$ in the production function:

$8(0) + 10K = 40$

$10K = 40$

$K = 4$

So, one solution is $L = 0$ and $K = 4$. The cost at this point is $C = 3(0) + 4(4) = 16$.

Second, set $K = 0$ in the production function:

$8L + 10(0) = 40$

$8L = 40$

$L = 5$

So, another solution is $L = 5$ and $K = 0$. The cost at this point is $C = 3(5) + 4(0) = 15$.

Comparing the costs, the minimum cost is 15, which occurs when $L = 5$ and $K = 0$.

3. Final Answer

L = 5; K = 0