A machine manufacturing company has a fixed monthly cost of $\$ 1000$ and direct costs of $\$ 8$ for each machine produced. The company estimates that 100 machines can be sold if the unit price is $\$ 40$, and that 10 more machines will be sold for each decrease of $\$ 2$ in the price. Find the price per unit that will maximize the profit.

Select one:
$\$ 60$
$\$ 34$
$\$ 22$
$\$ 15$
$\$ 7$

Step 1 ：Let's denote the following variables: p as the price per machine, n as the number of machines sold, R as the revenue, C as the costs, and P as the profit.

Step 2 ：We have the following relationships: \(n = 100 + 10 \times \frac{(40 - p)}{2}\), \(R = n \times p\), \(C = 1000 + 8 \times n\), and \(P = R - C\).

Step 3 ：We need to find the value of p that maximizes P. This can be done by finding the derivative of P with respect to p, setting it equal to zero, and solving for p.

Step 4 ：Let's write the equations in terms of p: \(n = 300 - 5p\), \(R = p \times (300 - 5p)\), \(C = 3400 - 40p\), and \(P = p \times (300 - 5p) + 40p - 3400\).

Step 5 ：Taking the derivative of P with respect to p, we get \(P' = 340 - 10p\).

Step 6 ：Setting the derivative equal to zero and solving for p, we get \(p = 34\).

Step 7 ：This means that the price per machine that maximizes the profit is $34.

Step 8 ：Final Answer: \(\boxed{34}\)