Pumps A and B each operate at a constant rate. A reservoir is initially empty. Pump B operates alone until it has filled of the reservoir, which takes minutes. During any equal-length time intervals, pump A delivers 4 gallons for every 3 gallons that pump B delivers. At that point, pump A is turned on while pump B continues operating. Which choice represents the total time, in minutes, from when pump B begins operating until the reservoir is full?
For a two-stage filling problem, first express one pump's rate using the given fraction of the reservoir and time. Use the rate ratio to find the other pump's rate, add the rates only during the stage when both pumps operate, and remember to add the time from the earlier stage at the end.
Hints
Compare the pump rates
Translate “4 gallons for every 3 gallons” into a multiplier relating pump A's rate to pump B's rate.
Use a fraction of the reservoir as a rate
Because pump B fills of the reservoir in minutes, express its rate in reservoirs per minute.
Account for both stages
Find how long both pumps need to fill the remaining , then include the time from the first stage.
Desmos Guide
Choose a convenient value for t
Algebra is faster for this question, but Desmos can verify the result. Set , which makes the fractional times easy to inspect.
Enter the rates
Enter for pump B's rate. Then enter for pump A's rate.
Calculate the total time
Enter . Desmos gives 23 minutes when , which corresponds to for any value of .
Step-by-step Explanation
Determine the rate relationship
For every 3 gallons pump B delivers, pump A delivers 4 gallons in the same time. Therefore, pump A's rate is of pump B's rate.
Write the combined filling rate
Pump B fills of the reservoir in minutes, so its rate is reservoir per minute.
Thus, pump A's rate is
Together, the pumps fill at a rate of
Fill the remaining part and add the first stage
After the first stage, of the reservoir remains. The time for both pumps to fill it is
Including the initial minutes when pump B operated alone gives