Suppose that a tank holds 100 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 40 minutes. Toricelli's Law gives the volume of water remaining in the tank after t minutes as
V(t)=100(1−t40)2
a.) Find V−1. What does V−1 represent?
b.) Find V−1(15). What does it represent?
a.) To find V−1, we set y=V(t)
y=100(1−t40)2Solve for t; divide by 100y100=(1−t40)2Take the square root1−t40=±√y100Add t40 and subtract √y100t40=1∓√y100Multiply by 40t=40(1∓√y100)Interchange t and yy=40(1∓√t100)Simplifyy=40(1∓√t10)
Thus, the inverse of V(t) is V−1(t)=40(1∓√t10).
If V(t) represents the volume of the water remaining in the tank, then V−1(t) represents the volume of the water drained from the tank.
b.) V−1(15)=40(1∓√1510)
V−1(15)=40+4√15 gallons and V−1(15)=40−4√15 gallons V−1(15) represents the volume of the water drained from the tank after 15 minutes. We choose V−1(15)=40+4√15 gallons because if we add the volume of the water remaining and drained from the tank after 15 minutes, the sum is close to 100 gallons.
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