Final answer:
The rate at which sugar-water solution is leaving the tank is 2 gal per minute. The rate at which the concentration of sugar in the tank is changing is 0, so the concentration of sugar in the tank remains constant. Therefore, the function x(t) for the amount of sugar in the tank is a constant function.
Step-by-step explanation:
To find a function for the amount of sugar in the tank, we need to consider the rate at which pure water is pouring into the tank and the rate at which the sugar-water solution is leaving the tank. Let's first determine the rate at which sugar-water solution is leaving the tank. Since the volume of the sugar-water solution in the tank remains constant, the rate at which the solution is leaving must be equal to the rate at which pure water is pouring in. So, the rate at which sugar-water solution is leaving the tank is 2 gal per minute.
Next, we need to find the rate at which the concentration of sugar in the tank is changing. The concentration of sugar in the tank is given as 2 lb of sugar for each gal of water. Since the volume of the tank remains constant and the concentration of sugar is changing, we can use the equation:
Rate of change of concentration of sugar = (Rate of sugar-water solution leaving tank - Rate of pure water pouring in) / Volume of tank
Plugging in the values, we get:
Rate of change of concentration of sugar = (2 - 2) / 50 = 0
Since the rate of change of concentration of sugar is 0, the concentration of sugar in the tank remains constant. Therefore, the function x(t) for the amount of sugar in the tank is a constant function.