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

Answered by imjagoll last updated on 24/Jun/21

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

$$\mathrm{we}\:\mathrm{convert}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{2t}+\mathrm{3}\:\mathrm{liters} \\ $$$$\mathrm{per}\:\mathrm{minute}\:\mathrm{to}\:\mathrm{60}\left(\mathrm{2t}+\mathrm{3}\right)=\mathrm{120t}+\mathrm{180} \\ $$$$\mathrm{liters}\:\mathrm{per}\:\mathrm{hours}.\:\mathrm{The}\:\mathrm{total}\:\mathrm{amount} \\ $$$$\mathrm{of}\:\mathrm{water}\:\mathrm{in}\:\mathrm{the}\:\mathrm{tank}\:\mathrm{at}\:\mathrm{time}\:\mathrm{T} \\ $$$$\mathrm{hours}\:\mathrm{past}\:\mathrm{noon}\:\mathrm{is}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{T}} \left(\mathrm{120t}+\mathrm{180}\right)\mathrm{dt}\:=\:\mathrm{60T}^{\mathrm{2}} +\mathrm{180T} \\ $$$$\mathrm{The}\:\mathrm{tank}\:\mathrm{is}\:\mathrm{full}\:\mathrm{when}\: \\ $$$$\:\:\:\mathrm{60T}^{\mathrm{2}} +\mathrm{180T}\:=\mathrm{1000} \\ $$$$\mathrm{solving}\:\mathrm{for}\:\mathrm{T}\:\mathrm{by}\:\mathrm{the}\:\mathrm{quadratic} \\ $$$$\mathrm{formula}\:\mathrm{we}\:\mathrm{get}\:\mathrm{T}\:\approx\:\mathrm{2}.\mathrm{849}\:\mathrm{hours} \\ $$$$\:\mathrm{past}\:\mathrm{noon},\:\mathrm{so}\:\mathrm{the}\:\mathrm{tank}\:\mathrm{is}\:\mathrm{full} \\ $$$$\:\mathrm{at}\:\mathrm{2}:\mathrm{51}\:\mathrm{pm}.\: \\ $$

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