Answer:
Explanation:
Height h = 1.03m
Volume v = 3780 gallons = 3780 * 0.0037851m^3 = 14.3073m^3
Time t = 13.5 mins = 13.5 * 60 = 810 seconds
Length of pool L = 14 inch = 14 * 2.54 = 35.56cm
width of pool b = 48 inch = 48 * 2.54 = 121.92 cm
a.) Consider the bernoulli's equation is given as:
![P_1+\rho gh_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho gh_2 + \frac{1}{2}\rho v_2^2 ...(1)](https://tex.z-dn.net/?f=P_1%2B%5Crho%20gh_1%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Crho%20v_1%5E2%20%3D%20P_2%20%2B%20%5Crho%20gh_2%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Crho%20v_2%5E2%20...%281%29)
consider the equation of bernoulli at the top of the pool
![P_0+\rho gh_1 + \frac{1}{2}\rho v_1^2 =constant ...(2)](https://tex.z-dn.net/?f=P_0%2B%5Crho%20gh_1%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Crho%20v_1%5E2%20%3Dconstant%20...%282%29)
where
atm pressure
At the top of the pool
, substitute in V_1 in equation (2)
![P_0+\rho gh_1 =constant ...(3)](https://tex.z-dn.net/?f=P_0%2B%5Crho%20gh_1%20%3Dconstant%20...%283%29)
Hence equation (3) serves as the bernoullis equation at the top.
b.) Consider the equation of bernoulli's at the opening of the pool
![P_2+\rho gh_2 + \frac{1}{2}\rho v_2^2 =constant ...(4)\\P_0+\rho gh_2 + \frac{1}{2}\rho v_2^2 =constant ...(5)](https://tex.z-dn.net/?f=P_2%2B%5Crho%20gh_2%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Crho%20v_2%5E2%20%3Dconstant%20...%284%29%5C%5CP_0%2B%5Crho%20gh_2%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Crho%20v_2%5E2%20%3Dconstant%20...%285%29)
where
atm pressure and ![h_2=0m](https://tex.z-dn.net/?f=h_2%3D0m)
![P_0+\rho v_1^2 =constant ...(6)](https://tex.z-dn.net/?f=P_0%2B%5Crho%20v_1%5E2%20%3Dconstant%20...%286%29)
Hence equation (6) serves as the bernoullis equation of water at the opening of the pool.
c.) Consider the equation (3) and (4)
Hence velocity is ![v_2=(\sqrt{2gh_1})m/s](https://tex.z-dn.net/?f=v_2%3D%28%5Csqrt%7B2gh_1%7D%29m%2Fs)
d.) consider (7)
![v_2=(\sqrt{2(9.81)(1.03)})=4.4954m/s(approx)](https://tex.z-dn.net/?f=v_2%3D%28%5Csqrt%7B2%289.81%29%281.03%29%7D%29%3D4.4954m%2Fs%28approx%29)
This is the norminal value of velocity
e.) consider the equation of flow rate interval of v and t
flow(t)=
hence this is the flow rate
f.) Consider the equation cross sectional area in terms of V,v2 and t
![AV_2=\frac{v}{t}\\\\A=\frac{v}{v_2t}(m^2)...8](https://tex.z-dn.net/?f=AV_2%3D%5Cfrac%7Bv%7D%7Bt%7D%5C%5C%5C%5CA%3D%5Cfrac%7Bv%7D%7Bv_2t%7D%28m%5E2%29...8)
hence this serves as the cross sectional area.
g.) Consider the equation of area from equation (8)
![A=\frac{v}{v_2t}\\=\frac{14.3073}{4.4954\times 810}=0.003929=0.00393m^2=39.3cm^2](https://tex.z-dn.net/?f=A%3D%5Cfrac%7Bv%7D%7Bv_2t%7D%5C%5C%3D%5Cfrac%7B14.3073%7D%7B4.4954%5Ctimes%20810%7D%3D0.003929%3D0.00393m%5E2%3D39.3cm%5E2)