Related Rates Calculus Problem?

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2 Answers

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  • 4 weeks ago

    let constant rate = c

    dV/dt = c - 6400

    radius to height relationship 

    2r/h = 6/8

    ∴ r = ⅜h

    Volume and volumetric rate

    V = ⅓πr²h

    V = ⅓π(⅜h)²(h)

    V = 3πh³/64

    dV/dt = ( 9πh²/64 )dh/dt

    c - 6400 = ( 9πh²/64 )dh/dt

    c - 6400 = ( 9π(500)²/64 )(18)

    c = ( 9π(500)²/64 )(18) + 6400

    c = 1994439 cm³/min

  • Everything is measured in cm

    dV/dt = x - 6400

    V = (pi/3) * r^2 * h

    r = 300 when h = 800 and r = 0 when h = 0.  r = (3/8) * h

    h = 500 , dh/dt = 18

    V = (pi/3) * r^2 * h

    V = (pi/3) * ((3/8) * h)^2 * h

    V = (pi/3) * (9/64) * h^3

    V = (3pi/64) * h^3

    dV/dt = (9pi/64) * h^2 * dh/dt

    x - 6400 = (9pi/64) * 500^2 * 18

    x = 6400 + (9pi/64) * 500^2 * 18

    x = 6400 + (9pi/64) * 5^2 * 100^2 * 2 * 9

    x = 6400 + (9pi/64) * 5^2 * (2^2 * 5^2)^2 * 2 * 9

    x = 6400 + (9pi/64) * 25 * 25^2 * 2^5 * 9

    x = 6400 + (9pi/2) * 25^3 * 9

    x = (12800 + 9 * 9 * 25^3 * pi) / 2

    x = (12800 + 9 * 5^3 * 9 * 5^3 * pi) / 2

    x = (12800 + (9 * 125)^2 * pi) / 2

    x = (12800 + 1125^2 * pi) / 2

    1125^2 =>

    (1000 + 125)^2 =>

    1000000 + 250000 + 15625 =>

    1265625

    x = (12800 + 1265625 * pi) / 2

    x = ‭1,994,439.1010997910337146415159816‬

    1994439 cubic cm per minute, roughly

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