To find the cost of the large box of assorted drinks, we need to multiply the price of each drink bottle by the total number of bottles in the box.
The price of each drink bottle is: \[ 1.75 \text{ dollars} \]
The number of bottles in the box is: \[ 25 \]
Now, we calculate the total cost by multiplying the price per bottle by the number of bottles: \[ \text{Total cost} = 1.75 \times 25 \]
Calculating this: \[ 1.75 \times 25 = 43.75 \]
Thus, the cost of the large box is: \[ \boxed{43.75} \]