Certainly! Let's break down the problem and find the different equivalent rates step-by-step.
### Given Data
- A shower drain can drain water at the rate of 480 gallons in [tex]\(\frac{2}{3}\)[/tex] hour.
### Step 1: Calculate the Rate in Gallons per Hour
We need to find out how many gallons it drains in one hour. Since the drain rate is given for [tex]\(\frac{2}{3}\)[/tex] hour, we can use the following calculation:
[tex]\[ \text{Rate in gallons per hour} = \frac{\text{Total gallons}}{\text{Time in hours}} \][/tex]
Plugging in the given values:
[tex]\[ \text{Rate in gallons per hour} = \frac{480 \text{ gallons}}{\frac{2}{3} \text{ hour}} = 720 \text{ gallons per hour} \][/tex]
So, the drain rate is [tex]\(720 \text{ gallons per hour}\)[/tex].
### Step 2: Convert the Rate to Gallons per Minute
Next, we need to find the rate in gallons per minute. There are 60 minutes in an hour, so we divide the rate in gallons per hour by 60:
[tex]\[ \text{Rate in gallons per minute} = \frac{720 \text{ gallons per hour}}{60 \text{ minutes per hour}} = 12 \text{ gallons per minute} \][/tex]
### Step 3: Convert the Rate to Gallons per Second
Now, let's convert the rate from gallons per minute to gallons per second. There are 60 seconds in a minute, so we divide the rate in gallons per minute by 60:
[tex]\[ \text{Rate in gallons per second} = \frac{12 \text{ gallons per minute}}{60 \text{ seconds per minute}} = 0.2 \text{ gallons per second} \][/tex]
### Step 4: Convert the Rate to Seconds per Gallon
Finally, we'll find how many seconds it takes to drain one gallon of water. We take the reciprocal of the rate in gallons per second:
[tex]\[ \text{Rate in seconds per gallon} = \frac{1 \text{ second}}{0.2 \text{ gallons}} = 5 \text{ seconds per gallon} \][/tex]
### Summary of Equivalent Rates
1. The drain rate is [tex]\(720 \text{ gallons per hour}\)[/tex].
2. The drain rate is [tex]\(12 \text{ gallons per minute}\)[/tex].
3. The drain rate is [tex]\(0.2 \text{ gallons per second}\)[/tex].
4. The drain rate is [tex]\(5 \text{ seconds per gallon}\)[/tex].
Each of these rates is equivalent to the given rate of draining 480 gallons in [tex]\(\frac{2}{3}\)[/tex] hour. The unit rate in gallons per minute is [tex]\(12 \text{ gallons per minute}\)[/tex].
