From the given information, we can calculate the total carrying costs, the number of times the company restocks per year, the total restocking costs, the $EOQ$, and the total cost.

The total carrying cost can be calculated as the product of the carrying cost per pair per year and the number of pairs in stock, which is $100$. Therefore, the total carrying cost is:[2]
$$\text{Carrying stock} = \text{((Initial + Ending)} / 2 ) = (100 + 0) / 2 = 50\text{ pairs}$$

$$\text{Total carrying cost} = 3 \times 50 = \$150$$

To find the number of times the company restocks per year, we can divide the total number of pairs sold per year by the number of pairs ordered each time. Since the company starts with $100$ pairs in stock and sells $600$ pairs per year, company has to order $600/100 = 6$ times.

The total restocking cost can be calculated as the product of the restocking cost per order and the number of orders placed per year. Since the company restocks 6 times per year, it places one order per year and a partial order every other year[3]. Therefore, the total restocking cost is:
$$\text{Total restocking cost} = $20/\text{order} \times 6 \text{ order/year} = $120/\text{year}$$

The $EOQ$ (Economic Order Quantity) can be calculated using the formula:
$$EOQ = \sqrt{(2DS)/H}$$

where $D$ is the annual demand, $S$ is the setup cost per order, and $H$ is the holding cost per unit per year. In this case, $D = 600$ pairs/year, $S =$ \$$20$/order, and $H =$ $$3$/pair/year. Therefore, the $EOQ$ is:

$$EOQ = \sqrt{(2 \times 600 \times $20)/$3} = 89.44\text{ pairs}$$

The total cost can be calculated as the sum of the carrying cost and the restocking cost. Since Thiewes sells 600 pairs per year, it will restock 600/89.44 = 6.7084 times.

The total restocking costs will be \$$20 \times 6.71 =$ \$$134.16.$

Then , the average inventory will be $89.44/2 = 44.72.$

The carrying costs will be \$$3 \times 44.72 =$ \$$134.16$, the same as the restocking costs.

Total cost = $134.16 + 134.16 = 268.32$.

Therefore, the total carrying costs for the hiking boots are \$$150$/year, the company restocks \$$6$ times per year, the total restocking costs are $120$year, the $EOQ$ is $89.44$ pairs, and the total cost is $$268.32$year.