Choice B is correct. It’s given that the shop’s inventory starts with $4,500$ paper cups and that the manager estimates that $70$ of these paper cups are used each day. Let $x$ represent the number of days in which the estimated supply of paper cups will reach $1,700$ . The equation $4,500 - 70 x = 1,700$ represents this situation. Subtracting $4,500$ from both sides of this equation yields $- 70 x = - 2,800$ . Dividing both sides of this equation by $negative 70$ yields $x equals 40$ . Therefore, based on this estimate, the supply of paper cups will reach $1,700$ in $40$ days.

Choice A is incorrect. After $20$ days, the estimated supply of paper cups would be $4,500 minus 70 left parenthesis 20 right parenthesis$ , or $3,100$ cups, not $1,700$ cups.

Choice C is incorrect. After $60$ days, the estimated supply of paper cups would be $4,500 minus 70 left parenthesis 60 right parenthesis$ , or $300$ cups, not $1,700$ cups.

Choice D is incorrect. After $80$ days, the estimated supply of paper cups would be $4,500 minus 70 left parenthesis 80 right parenthesis$ , or $negative 1,100$ cups, which isn't possible.