JCLEONES wrote:
Reserve tank 1 is capable of holding z gallons of water. Water is pumped into tank 1, which starts off empty at a rate of x gallons per minute. Tank 1 simultaneously leaks water at a rate of y gallons per minute (x > y).The water that leaks out of tank 1 drips into tank 2, which also starts off empty. If the total capacity of tank 2 is twice the number of gallons that remains in tank 1 after 1 minute, does tank 1 fill up before tank 2?
(1) \(zy < 2x^2-4xy+2y^2\)
(2) Total capacity of tank 2 is less than one half that of tank 1.
Tank 1:
Since water is PUMPED IN at x gallons per minute but LEAKS OUT at y gallons per minute, the net gain per minute = x-y.
Since the z-gallon tank is filled at a net rate of x-y gallons per minute, we get:
Time fill tank 1 \(= \frac{capacity}{rate} = \frac{z}{x-y}\)
Tank 2:
The total capacity of tank 2 is twice the number of gallons that remains in tank 1 after 1 minute.After 1 minute, the number of gallons in tank 1 = (net gain per minute)(one minute) = (x-y)(1) = x-y.
Since the capacity of tank 2 is twice this number of gallons, we get:
Capacity of tank 2 = 2(x-y) = 2x-2y.
The water that leaks out of tank 1 drips into tank 2.
Since water leaks from tank 1 into tank 2 at a rate of y gallons per minute, we get:
Time to fill tank 2 \(= \frac{capacity}{rate} = \frac{2x - 2y}{y}\)
Does tank 1 fill up before tank 2?In other words:
Is the time fill tank 1 less than the time to fill tank 2?Original question stem:
Is \(\frac{z}{x-y}< \frac{2x-2y}{y}\)?
Simplifying the question stem, we get:
\(zy < (2x-2y)(x-y)\)
\(zy < 2x^2-4xy+2y^2\)
Question stem, rephrased:
Is \(zy < 2x^2-4xy+2y^2\)?
Statement 1: \(zy < 2x^2-4xy+2y^2\)
The answer to the rephrased question stem is YES.
SUFFICIENT.
Statement 2:
Since the capacity of tank 2 = 2x-2y and the capacity of tank 1 = z, we get:
\(2x-2y < \frac{1}{2}z\)
\(4x-4y < z\)
\(4 > \frac{z}{x-y}\)
\(\frac{z}{x-y} < 4\)
No way to answer the original question stem or the rephrased question stem.
INSUFFICIENT.
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