Respuesta :
Answer:
The time it will take is approximately 104,128 seconds
Step-by-step explanation:
The given parameters are;
The diameter of the tank, D = 24 ft.
Therefore, the tank radius, R = D/2 = 24 ft./2 = 12 ft.
The height of the tower on which the tank sits, h = 60 ft.
The hose through which the tank is filled is attached at the bottom of the tank
The radius of a slice of sphere, r = √(R² - y²) = √(12² - y²)
The power of the pump which is used to deliver the water = 1.5 horsepower
The volume of a slice of water in the tank, V = π·r²·Δx = π·(√(12² - y²))²·Δx ft³ = (144 - y²)·π·Δx ft³
The force of the slice, F = V·g·ρ = (144 - x²)·π·Δx ft³ × 62.5 lb/ft³
Let '[tex]y_i[/tex]' represent the height to which each slice is pumped in the tank, we have;
y = R - (√(R² - x²)) = 12 - (√(144 - x²)
[tex]\lim\limits_{n \to \infty}\ \sum\limits_{i=0}^n (144 - x^2) \cdot \pi \times 62.5 \cdot y_i\ \Delta x \ lb[/tex]
The work done is therefore;
[tex]W = 62.5\times\pi \times \int\limits^{12}_{-12} {(144 - x^2) \cdot(12-(\sqrt{144-x^2} ) \ } \, dx = 632044.366475 \ ft. \cdot lb[/tex]
The work done in filling the tank, W = 632,044.366475 ft·lb
The work done in lifting the water to the base of the tank, W₂ = V·ρ·g×h
[tex]\therefore W_2 = 60 \times 62.5\times\pi \times \int\limits^{12}_{-12} {(144 - x^2) \ } \, dx = 632044.366475 \ ft. \cdot lb = 85273382.0257[/tex]
Therefore, W₂ = 85,273,382.0257 ft.lb
The total work done by the pump,
W = 85,273,382.0257 ft.lb + 632,044.366475 ft·lb = 85,905,426.3922 ft.lb
The time it will take the pump to fill the tank, 't', is given as follows;
1
Power, P = Work, W/(Time, t)
∴ t = W/P
P = 1.5 HP = 550 × 1.5 ft·lb/s = 825 ft·lb/s
t = 85,905,426.3922 ft.lb/(825 ft·lb/s) = 104,127.789566 s
The time it will take, t ≈ 104,128 seconds
