Let's solve the problem step-by-step using the given data. We know the prices of labor and capital are [tex]$9 and $[/tex]15, respectively. We need to determine the optimal quantities of labor and capital to minimize costs and then find the total cost.
Step 1: Identify the optimal quantities of labor and capital.
From the given data:
- The optimal quantity of labor is 5 units.
- The optimal quantity of capital is 2 units.
Step 2: Calculate the cost incurred by using these quantities.
- Labor Cost:
- Price of labor = [tex]$9 per unit
- Quantity of labor = 5 units
- Total cost of labor = \( 9 \times 5 = \$[/tex]45 \)
- Capital Cost:
- Price of capital = [tex]$15 per unit
- Quantity of capital = 2 units
- Total cost of capital = \( 15 \times 2 = \$[/tex]30 \)
Step 3: Sum the costs to find the total cost.
- Total cost = Total cost of labor + Total cost of capital
- Total cost = [tex]\( \$45 + \$30 = \$75 \)[/tex]
Thus, the firm's total cost at the profit-maximizing level of output is \[tex]$75.
Therefore, the correct answer from the given options (listed as $[/tex]\[tex]$ 90$[/tex], [tex]$\$[/tex] 47[tex]$, $[/tex]\[tex]$ 126$[/tex], [tex]$\$[/tex] 106[tex]$) is none of these correctly represent the total cost calculated. However, based on calculated results, the correct total cost is \$[/tex]75.
