Wednesday, January 18, 2017

College Algebra, Chapter 3, 3.7, Section 3.7, Problem 76

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(1t40)2

a.) Find V1. What does V1 represent?

b.) Find V1(15). What does it represent?



a.) To find V1, we set y=V(t)


y=100(1t40)2Solve for t; divide by 100y100=(1t40)2Take the square root1t40=±y100Add t40 and subtract y100t40=1y100Multiply by 40t=40(1y100)Interchange t and yy=40(1t100)Simplifyy=40(1t10)


Thus, the inverse of V(t) is V1(t)=40(1t10).

If V(t) represents the volume of the water remaining in the tank, then V1(t) represents the volume of the water drained from the tank.

b.) V1(15)=40(11510)

V1(15)=40+415 gallons and V1(15)=40415 gallons V1(15) represents the volume of the water drained from the tank after 15 minutes. We choose V1(15)=40+415 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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