The time it takes to fill a water tank varies inversely with the water rate of the hose. At 8 gallons per minute, a hose can fill the tank in 45 minutes. How long will it take to fill the same tank at 15 gallons per minute?

Step 1 ：Given that the time it takes to fill a tank varies inversely with the rate of the hose, we can use the formula for inverse variation, which is \(y = k/x\), where \(y\) is the time it takes to fill the tank, \(x\) is the rate of the hose, and \(k\) is the constant of variation.

Step 2 ：We are given that at 8 gallons per minute, a hose can fill the tank in 45 minutes. We can plug these values into the formula to find the constant of variation: \(45 = k/8\). Solving for \(k\), we find that \(k = 360\).

Step 3 ：Now we can use the constant of variation to find the time it takes to fill the tank at 15 gallons per minute. Plugging \(k = 360\) and \(x = 15\) into the formula, we get \(y = 360/15\).

Step 4 ：Solving for \(y\), we find that it will take \(\boxed{24}\) minutes to fill the tank at 15 gallons per minute.