Using the inverse variation formula, we have:
\(T_1 \cdot R_1 = T_2 \cdot R_2\)
Where:
\(T_1 = \text{time required to empty the tank at the rate of } 600 \text{ gallons per minute}\)
\(R_1 = 600 \text{ gallons per minute}\)
\(T_2 = \text{time required to empty the tank at the rate of } 1000 \text{ gallons per minute}\)
\(R_2 = 1000 \text{ gallons per minute}\)
Plugging in the values we have:
\(45 \cdot 600 = T_2 \cdot 1000\)
\(27000 = 1000T_2\)
\(T_2 = \frac{27000}{1000}\)
\(T_2 = 27\)
Therefore, it will take the pump 27 minutes to empty the tank at the rate of 1000 gallons per minute.