Choice A is correct. It’s given that the truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than $\mathrm{4,600}$ pounds. If the trailer has a weight of $500$ pounds and each box weighs $120$ pounds, the expression $500+120b$, where $b$ is the number of boxes, gives the combined weight of the trailer and the boxes. Since the combined weight must be no more than $\mathrm{4,600}$ pounds, the possible numbers of boxes the truck can tow are given by the inequality $500+120b\le \mathrm{4,600}$. Subtracting $500$ from both sides of this inequality yields $120b\le \mathrm{4,100}$. Dividing both sides of this inequality by $120$ yields $b\le \frac{205}{6}$, or $b$ is less than or equal to approximately $34.17$. Since the number of boxes, $b$, must be a whole number, the maximum number of boxes the truck can tow is the greatest whole number less than $34.17$, which is $34$.

Choice B is incorrect. Towing the trailer and $35$ boxes would yield a combined weight of $\mathrm{4,700}$ pounds, which is greater than $\mathrm{4,600}$ pounds.

Choice C is incorrect. Towing the trailer and $38$ boxes would yield a combined weight of $\mathrm{5,060}$ pounds, which is greater than $\mathrm{4,600}$ pounds.

Choice D is incorrect. Towing the trailer and $39$ boxes would yield a combined weight of $\mathrm{5,180}$ pounds, which is greater than $\mathrm{4,600}$ pounds.