Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A spherical water tank, 24

Question

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A spherical water tank, 24 ft in in diameter, sits atop a 60 ft tower. The tank is filled by a hose attached to the bottom of the sphere. If a 1.5 horsepower pump is used to deliver water up to the tank, how long

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Hồng Cúc 4 years 2021-07-23T20:08:04+00:00 1 Answers 27 views 0

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  1. trucchi
    0
    2021-07-23T20:09:34+00:00
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    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 ‘y_i‘ represent the height to which each slice is pumped in  the tank, we have;

    y = R – (√(R² – x²)) = 12 – (√(144 – x²)

    \lim\limits_{n \to \infty}\ \sum\limits_{i=0}^n (144 - x^2) \cdot \pi \times 62.5 \cdot y_i\ \Delta x \ lb

    The work done is therefore;

    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

    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

    \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

    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

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