Torricelli's Law provides the volume V of water still in the tank after t minutes if a tank holds 5000 gallons of water and the water drains from the bottom of the tank in 40 minutes.(1−t402)5000=V0≤t≤40.Find the rate at which the tank's water is emptying after (a) five minutes and (b) ten minutes. At what time is the water flowing out the fastest? The slowest?

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Torricelli&#039;s Law provides the volume V of water still in the tank after t minutes if a tank holds 5000 gallons of water and the water drains from the bottom of the tank in 40 minutes.(1−t402)5000=V0≤t≤40.Find the rate at which the tank&#039;s water is emptying after (a) five minutes and (b) ten minutes. At what time is the water flowing out the fastest? The slowest?

Caren · Accepted Answer

Torricelli&#039;s Law provides the volume V of water still in the tank after t minutes if a tank holds 5000 gallons of water and the water drains from the bottom of the tank in 40 minutes.(1−t402)5000=V0≤t≤40IS IT&amp;nbsp;T⇐40?Find the rate at which water is draining from the tank after (a) 5min, (b) 10 min. At what time is the water flowing out the fastest? The slowest?WATER FLOWN OUT IN TIME&amp;nbsp;T=W=5000−5000[1−T2/40]DW/DT =−5000(−2T)/40=250TAT T=5..RATE OF WATER OUT FLOW=DW/DT&amp;nbsp;=250∗5=1250AT T=10DW/DT=250∗10=2500&amp;nbsp;GPMAS PER THE GIVEN EQN.WATER LOW RATE (=250T)IS INCREASING WITH T AND HENCE HIGHEST AT&amp;nbsp;T=40&amp;nbsp;AND T=0BUT THIS APPEARS PHYSICALLY FAULTY AND THE GIVEN EQN. APPEARS TO BE DOUBT FULL

Jeffrey Jordon · Accepted Answer

The rate at which water is draining from the tank is the derivative of V(t). Use the chain rule to find dVdt.V(t)=5000(1−140t)2Chain rule : ddx(f(g(x)))=f′(g(x))g′(x)f(x)=5000x2,g(x)=1−x40f′(x)=2(5000)x,g′(x)=−140dVdt=2(5000)(1−140t)(−140)=−250(1−140t)Step 2Plug in t=5 to find the flow rate at that point in time. The volume of water in the tank decreases by this amount, so the flow rate must have this magnitude (the negative sign isn&#039;t necessary on the answer).dVdt=−250(1−140(5))=−218.75 gallonsmin218.75 gallonsmin

karton · Accepted Answer

Result:(a) The rate at which the tank&#039;s water is emptying after five minutes is −1250 gallons per minute.(b) The rate at which the tank&#039;s water is emptying after ten minutes is −2500 gallons per minute.(c) The water is flowing out the fastest at time t = 0 minutes.(d) The water is flowing out the slowest at time t = 40 minutes.Solution:First, let&#039;s find the volume V of water still in the tank after t minutes using Torricelli&#039;s Law equation:(1−t240)5000=VNow, let&#039;s calculate the rate at which the tank&#039;s water is emptying after five minutes:To find the rate at a specific time, we need to differentiate the volume equation with respect to time t:dVdt=−10000t40Substituting t = 5 into the equation:dVdt=−10000(5)40=−1250&amp;nbsp;gallons&amp;nbsp;per&amp;nbsp;minuteTherefore, the rate at which the tank&#039;s water is emptying after five minutes is 1250 gallons per minute (negative sign indicates emptying).Now, let&#039;s calculate the rate at which the tank&#039;s water is emptying after ten minutes:Substituting t = 10 into the derivative equation:dVdt=−10000(10)40=−2500&amp;nbsp;gallons&amp;nbsp;per&amp;nbsp;minuteSo, the rate at which the tank&#039;s water is emptying after ten minutes is 2500 gallons per minute (negative sign indicates emptying).Next, we need to find the time at which the water is flowing out the fastest. To do this, we set the derivative equation equal to zero and solve for t:−10000t40=0Simplifying the equation, we find:t=0Therefore, at time t = 0 minutes, the water is flowing out the fastest.Finally, we need to find the time at which the water is flowing out the slowest. Again, we set the derivative equation equal to zero and solve for t:−10000t40=0Simplifying the equation, we find:t=40Therefore, at time t = 40 minutes, the water is flowing out the slowest.

star233 · Accepted Answer

Torricelli&#039;s Law states that the volume V of water remaining in the tank after t minutes can be determined using the equation:(1−t240)5000=V,0≤t≤40.To find the rate at which the tank&#039;s water is emptying, we differentiate the equation with respect to t:dVdt=−t20·5000.(a) To find the rate after five minutes (t=5):dVdt|t=5=−520·5000=−1250.Therefore, the tank&#039;s water is emptying at a rate of −1250 gallons per minute after five minutes.(b) To find the rate after ten minutes (t=10):dVdt|t=10=−1020·5000=−2500.Thus, the tank&#039;s water is emptying at a rate of −2500 gallons per minute after ten minutes.To determine when the water is flowing out the fastest, we need to find the time t that maximizes |dVdt|. Since dVdt is a linear function, it decreases uniformly from its initial value. Therefore, the water is flowing out the fastest at t=0 (when the tank is full) and the slowest at t=40 (when the tank is empty).

If a tank holds 5000 gallons of water, which drains from the bttomof the tank in 40 minutes, then torricelli's Law gives the volume Vof water remainin

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