The correct answer is $\boxed{\text{B}}$.
The economic order quantity (EOQ) is the optimal quantity of an item to order at a time. It is the quantity that minimizes the total cost of ordering and carrying inventory. The EOQ is calculated as follows:
$$EOQ = \sqrt{\frac{2DC}{h}}$$
where:
- $D$ is the annual demand for the item, in units
- $C$ is the cost of placing an order, in dollars
- $h$ is the holding cost per unit per year, in dollars
In this case, we are given that $D = 1500$ units, $C = 10$ dollars, and $h = 0.1$ dollars. Substituting these values into the EOQ formula, we get:
$$EOQ = \sqrt{\frac{2 \times 1500 \times 10}{0.1}} = 15000$$
This means that the optimal quantity to order at a time is 15,000 units.
The number of deliveries in a year is equal to the annual demand divided by the EOQ. In this case, the annual demand is 1500 units and the EOQ is 15,000 units. Therefore, the number of deliveries in a year is:
$$\frac{1500}{15000} = 0.1 = \boxed{10}$$
Option A is incorrect because it is the number of deliveries in a year if the EOQ is 16,000 units. Option C is incorrect because it is the number of deliveries in a year if the EOQ is 10,000 units. Option D is incorrect because it is the number of deliveries in a year if the EOQ is 14,000 units.