Question 41·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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Pumps A and B each operate at a constant rate. A reservoir is initially empty. Pump B operates alone until it has filled $\frac{2}{5}$ of the reservoir, which takes $t$ 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?

A work-rate problem with one pump starting later has two intervals: one pump alone, then both pumps together. Measure output in fractions of the reservoir per $t$ -minute block, so you don't need its capacity in gallons. Add the pumps' outputs during the second interval, use a Desmos regression to find its length, then add the first interval. The trap is reporting only the time after the second pump starts.

Hints

Hint 1
A rate tells you how much is filled in a given time. Use one $t$ -minute stretch as a block: what fraction of the reservoir does B fill in one block?
Hint 2
The 4-for-3 comparison applies over equal-length times. If B fills a certain fraction in one block, A fills $\frac{4}{3}$ times that fraction. What do they fill together in one block?
Hint 3
Only $\frac{3}{5}$ remains when A starts. Find how many blocks both pumps need for that portion, then include the block B already spent working alone.

Step-by-step

Measure both pump rates per $t$ -minute block
Step 1
Find B's output per block
Call a $t$ -minute stretch one block. B fills $\frac{2}{5}$ of the reservoir in one block, so let $b=\frac{2}{5}$ be its output per block. Type $b=\frac{2}{5}$ ; Desmos shows $b=0.4$ . Using fractions of the reservoir means you don't need its capacity in gallons.
Step 2
Find A's output per block
The ratio compares outputs over the same length of time. A delivers 4 for every 3 B delivers, so A's output per block is $a=\frac{4b}{3}$ , not $\frac{3b}{4}$ . Type $a=\frac{4b}{3}$ ; Desmos shows $a\approx0.533333$ of the reservoir per block.
Step 3
Add the outputs while both pumps run
B continues operating after A starts, so their outputs in the same block add. Type $c=a+b$ for their combined output; Desmos shows $c\approx0.933333$ of the reservoir per block. Add outputs for pumps working at the same time, but add times for consecutive intervals.
Step 4
Find what remains when A starts
B has filled $\frac{2}{5}$ of one reservoir, so the unfilled portion is:
$1-\frac{2}{5}=\frac{3}{5}.$
That's the portion the pumps must fill together.
Step 5
Model the second interval
Let $x_1$ be the number of $t$ -minute blocks after A starts. Output per block times the number of blocks must equal the $\frac{3}{5}$ still unfilled:
$cx_1=\frac{3}{5}.$
Step 6
Solve for the time both pumps run
Type $cx_1\sim\frac{3}{5}$ . The $\sim$ tells Desmos to find the value of $x_1$ that makes the two sides equal. Under PARAMETERS, it reports $x_1\approx0.642857$ . That's only the second interval, measured in $t$ -minute blocks.
Step 7
Include B's head start
B worked alone for one full block before both pumps worked for $x_1$ blocks. Type $1+x_1$ ; Desmos shows about $1.642857$ , and its fraction button gives $\frac{23}{14}$ . Each block lasts $t$ minutes, so the total time from B's start is $\frac{23t}{14}$ minutes. Choice A.

Question 41·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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