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Nataly_w [17]
2 years ago
6

Water leaks from a vertical cylindrical tank through a small hole in its base at a volumetric rate proportional to the square ro

ot of the volume of water remaining. The tank initially contains 300 liters and 22 liters leak out during the first day A. When will the tank be half empty?t16 day (include units.) B. How much water will remain in the tank after 5 days? volume 198 L (include units)
Mathematics
1 answer:
zloy xaker [14]2 years ago
5 0

Answer:

After 7.84 the tank be half empty. The water remain in the tank after 5 days is 198.401 L.

Step-by-step explanation:

Consider the provided information.

It is given that a small hole in its base at a rate proportional to the square root of the volume of water remaining. The tank initially contains 300 liters and 22 liters leak out during the first day.

The rate of water leak can be written as:

\frac{dV}{dt}\propto \sqrt{V}

Let k be the constant of proportionality.

\frac{dV}{dt}=k \sqrt{V}

Integrate both the sides as shown:

\frac{dV}{\sqrt{V}}=k dt\\\int\frac{dV}{\sqrt{V}}=\int k dt\\2\sqrt{V} =kt+c

Since for t=0 the volume was 300.

2\sqrt{300} =k(0)+c\\20\sqrt{3} =c\\c=34.641

Now substitute the value of c in above equation.

2\sqrt{V} =kt+34.641

22 liters leak out during the first day, thus now the remaining volume is 300-22=278 liters.

2\sqrt{278} =k(1)+34.641\\33.347 =k+34.641\\k=33.347 -34.641\\k=-1.294

Thus, the required equation is:2\sqrt{V} =-1.294t+34.641

Part (A) When will the tank be half empty.

Substitute v=150 liters for half empty in above equation.

2\sqrt{150} =-1.294t+34.641

24.495 =-1.294t+34.641

-10.146 =-1.294t

t=7.84

Hence, after 7.84 the tank be half empty.

Part (B) How much water will remain in the tank after 5 days.

Substitute the value of t=5 in 2\sqrt{V} =-1.294t+34.641

2\sqrt{V} =-1.294(5)+34.641

2\sqrt{V} =28.171

\sqrt{V} =14.0855

V =198.401

Hence, the water remain in the tank after 5 days is 198.401 L.

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