Two reservoirs are connected by two parallel pipes of equal length and of diameters 20 cm and 10 cm, as shown in the figure (not drawn to scale). When the difference in the water levels of the reservoirs is 5 m, the ratio of discharge in the larger diameter pipe to the discharge in the smaller diameter pipe is \(\underline{\hspace{2cm}}\) (round off to two decimal places).
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The discharge ratio for pipes with different diameters is proportional to the square of their diameters.
% Solution
We will use the Darcy-Weisbach equation for discharge in each pipe:
\[
Q = \frac{\pi D^2}{4} \times \sqrt{\frac{2gH}{fL}}
\]
where:
- \( D \) is the diameter of the pipe,
- \( g \) is the acceleration due to gravity,
- \( H \) is the head difference (5 m),
- \( f \) is the friction factor,
- \( L \) is the length of the pipe (equal for both pipes).
The discharge is proportional to \( D^2 \), as friction factor and other factors are the same for both pipes. Hence, the ratio of discharges is:
\[
\frac{Q_{\text{large}}}{Q_{\text{small}}} = \left( \frac{D_{\text{large}}}{D_{\text{small}}} \right)^2 = \left( \frac{20}{10} \right)^2 = 4
\]
Thus, the ratio of discharges is \( \boxed{4.00} \).
