Question

When friction and contraction of the water at the hole are taken into account, the model in Problem 11 becomes\(\frac{{dh}}{{dt}} = - c\frac{{{A_h}}}{{{A_w}}}\sqrt {2gh} ,\)where \({\bf{0}} < {\bf{c}} < {\bf{1}}\)How long will it take the tank in Problem 11(b) to empty if\(c = 0.6\)? See Problem 13 in Exercises 1.3.

Short answer

The time at which the tank will be empty\( \approx 12.65{\rm{ minutes }}\)

Step-by-step solution

Define the equation of height of the water

The height of water in this tank is described by the\(\frac{{dh}}{{dt}} = - \frac{{{A_h}}}{{{A_w}}}\sqrt {2gh} \)

Where\({A_w}\)is the cross-sectional area of water, with the condition

obtain the amount of water in the cylinder at time t as the following:

Here \(\frac{{dh}}{{\sqrt {2gh} }} = - c\frac{{{A_h}}}{{{A_w}}}dt\)

\(\frac{1}{{{{(2gh)}^{\frac{1}{2}}}}}dh = - c\frac{{{A_h}}}{{{A_w}}}\int d t\)

\(\frac{1}{{2g \times 2}}{(2gh)^{\frac{1}{2}}} = - c\frac{{{A_h}}}{{{A_w}}}t + {c_1}\)

\({(2 \times 32h)^{\frac{1}{2}}} = - \frac{{128c{A_h}}}{{{A_w}}}t + {c_2}\)

\(8{h^{\frac{1}{2}}} = - \frac{{128c{A_h}}}{{{A_w}}}t + {c_2}\)

\({h^{\frac{1}{2}}} = - \frac{{16c{A_h}}}{{{A_w}}}t + {c_3}\)

Then we have, \(h(t) = {\left( { - 16c\frac{{{A_h}}}{{{A_w}}}t + {c_3}} \right)^2}\)……(1)

Find the value of constant \({C_3}\).

Substitute the condition \((h,t) = (H{\rm{ft}},0{\rm{ minutes }})\)

\(\begin{aligned}{l}H = {\left( {0 + {c_3}} \right)^2}\\{\left( {{c_3}} \right)^2} = H\end{aligned}\)

Then \({c_3} = \sqrt H \)

Now subnstitute in equation 1

\(h(t) = {\left( { - 16c\frac{{{A_h}}}{{{A_w}}}t + \sqrt H } \right)^2}\)………(2)

obtain the time at which the tank is empty\((h = 0){\rm{ with }}c = 0.6\)

Using eqution 2 we have

\(0 = {\left( { - \frac{{16 \times 0.6 \times \pi {{\left( {\frac{1}{{24}}} \right)}^2}}}{{\pi {{(2)}^2}}}t + \sqrt {10} } \right)^2}\)

\( - \frac{{16 \times 0.6}}{{4 \times 576}}t + \sqrt {10} = 0\)

\(\frac{1}{{240}}t = \sqrt {10} \)

\(t = 240 \times \sqrt {10} \)

\( = 758.95{\rm{ seconds }}\)

\( \approx 12.65{\rm{ minutes }}\)

The time at which the tank will be empty\( \approx 12.65{\rm{ minutes }}\)