Final answer:
The 18-month-old child presents with signs of hypovolemic shock due to severe vomiting and diarrhea, and the primary treatment involves fluid rehydration and circulation support.
Step-by-step explanation:
The primary problem for which you will treat this 18-month-old child, who presents with a history of fever, diarrhea, and vomiting, is hypovolemic shock likely resulting from fluid losses associated with severe vomiting and diarrhea. Key indicators for this diagnosis include the child's weak radial pulse compared to the carotid pulse, pale, cool, and clammy skin, the absence of tears when crying, and overall signs of dehydration and circulatory compromise. The immediate treatment will focus on stabilizing the child's condition, which includes administering fluids to counteract the dehydration and possibly using medications to support blood pressure if necessary.