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Question Number 144301 by liberty last updated on 24/Jun/21

Answered by imjagoll last updated on 24/Jun/21

$$\mathrm{we}\mathrm{convert}\mathrm{the}\mathrm{rate}\mathrm{of}2\mathrm{t}+3\mathrm{liters}$$$$\mathrm{per}\mathrm{minute}\mathrm{to}60(2\mathrm{t}+3)=120\mathrm{t}+180$$$$\mathrm{liters}\mathrm{per}\mathrm{hours}.\mathrm{The}\mathrm{total}\mathrm{amount}$$$$\mathrm{of}\mathrm{water}\mathrm{in}\mathrm{the}\mathrm{tank}\mathrm{at}\mathrm{time}\mathrm{T}$$$$\mathrm{hours}\mathrm{past}\mathrm{noon}\mathrm{is}\mathrm{the}\mathrm{integral}$$$${\int }_0^{\mathrm{T}}(120\mathrm{t}+180)\mathrm{dt}={60\mathrm{T}}^2+180\mathrm{T}$$$$\mathrm{The}\mathrm{tank}\mathrm{is}\mathrm{full}\mathrm{when}$$$${60\mathrm{T}}^2+180\mathrm{T}=1000$$$$\mathrm{solving}\mathrm{for}\mathrm{T}\mathrm{by}\mathrm{the}\mathrm{quadratic}$$$$\mathrm{formula}\mathrm{we}\mathrm{get}\mathrm{T}\approx 2.849\mathrm{hours}$$$$\mathrm{past}\mathrm{noon},\mathrm{so}\mathrm{the}\mathrm{tank}\mathrm{is}\mathrm{full}$$$$\mathrm{at}2:51\mathrm{pm}.$$

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