Answer :
To determine which tank is leaking faster, we need to calculate the leakage rate (in gallons per minute) for both tanks and then compare the rates.
Step-by-step Solution:
1. Identify the leakage amount and time for each tank:
- Tank [tex]$A$[/tex] leaks [tex]\(\frac{1}{16}\)[/tex] of a gallon in [tex]\(\frac{1}{12}\)[/tex] of a minute.
- Tank [tex]$B$[/tex] leaks [tex]\(\frac{3}{80}\)[/tex] of a gallon in [tex]\(\frac{1}{30}\)[/tex] of a minute.
2. Calculate the leakage rate for Tank [tex]$A$[/tex]:
The leakage rate is defined as the amount of water leaked per minute. To find this rate for Tank [tex]$A$[/tex], we divide the amount of water leaked by the time taken.
[tex]\[
\text{Leakage rate of Tank } A = \frac{\frac{1}{16} \text{ gallons}}{\frac{1}{12} \text{ minutes}} = \frac{1}{16} \times \frac{12}{1} = \frac{12}{16} = 0.75 \text{ gallons per minute}
\][/tex]
3. Calculate the leakage rate for Tank [tex]$B$[/tex]:
Similarly, for Tank [tex]$B$[/tex], we divide the amount of water leaked by the time taken.
[tex]\[
\text{Leakage rate of Tank } B = \frac{\frac{3}{80} \text{ gallons}}{\frac{1}{30} \text{ minutes}} = \frac{3}{80} \times \frac{30}{1} = \frac{3 \times 30}{80} = \frac{90}{80} = 1.125 \text{ gallons per minute}
\][/tex]
4. Compare the leakage rates:
- Tank [tex]$A$[/tex] has a leakage rate of [tex]\(0.75\)[/tex] gallons per minute.
- Tank [tex]$B$[/tex] has a leakage rate of [tex]\(1.125\)[/tex] gallons per minute.
Since [tex]\(1.125\)[/tex] gallons per minute is greater than [tex]\(0.75\)[/tex] gallons per minute, Tank [tex]$B$[/tex] is leaking faster than Tank [tex]$A$[/tex].
Conclusion:
Tank [tex]$B$[/tex] is leaking faster.
Step-by-step Solution:
1. Identify the leakage amount and time for each tank:
- Tank [tex]$A$[/tex] leaks [tex]\(\frac{1}{16}\)[/tex] of a gallon in [tex]\(\frac{1}{12}\)[/tex] of a minute.
- Tank [tex]$B$[/tex] leaks [tex]\(\frac{3}{80}\)[/tex] of a gallon in [tex]\(\frac{1}{30}\)[/tex] of a minute.
2. Calculate the leakage rate for Tank [tex]$A$[/tex]:
The leakage rate is defined as the amount of water leaked per minute. To find this rate for Tank [tex]$A$[/tex], we divide the amount of water leaked by the time taken.
[tex]\[
\text{Leakage rate of Tank } A = \frac{\frac{1}{16} \text{ gallons}}{\frac{1}{12} \text{ minutes}} = \frac{1}{16} \times \frac{12}{1} = \frac{12}{16} = 0.75 \text{ gallons per minute}
\][/tex]
3. Calculate the leakage rate for Tank [tex]$B$[/tex]:
Similarly, for Tank [tex]$B$[/tex], we divide the amount of water leaked by the time taken.
[tex]\[
\text{Leakage rate of Tank } B = \frac{\frac{3}{80} \text{ gallons}}{\frac{1}{30} \text{ minutes}} = \frac{3}{80} \times \frac{30}{1} = \frac{3 \times 30}{80} = \frac{90}{80} = 1.125 \text{ gallons per minute}
\][/tex]
4. Compare the leakage rates:
- Tank [tex]$A$[/tex] has a leakage rate of [tex]\(0.75\)[/tex] gallons per minute.
- Tank [tex]$B$[/tex] has a leakage rate of [tex]\(1.125\)[/tex] gallons per minute.
Since [tex]\(1.125\)[/tex] gallons per minute is greater than [tex]\(0.75\)[/tex] gallons per minute, Tank [tex]$B$[/tex] is leaking faster than Tank [tex]$A$[/tex].
Conclusion:
Tank [tex]$B$[/tex] is leaking faster.
