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Math Help - A Question about cylinders

  1. #1
    sharlen
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    A Question about cylinders

    This will probably be quite simple to work out, but my maths A Level was 14 years ago, and it's all a bit vague now.

    We have a petrol tank (a cylinder) which is 5280mm long and orientated horizontally (i.e. on its side). The volume of this tank is 15,000 Ltrs.

    We need to fill this tank, BUT leaving a 250mm air gap at the top.

    Obviously if the tank were orientated vertically it makes life easier (15,000 divided by 5280 = the litres per millimetre x 250mm = the litres occupying a 250mm gap. Deduct this from the 15,000 = the amount of fuel to put into the tank leaving a 250mm gap.).

    The tank is however on its side with the filling cap centrally located. We therefore have the incremental curvature of the tank to consider and we are at a loss to how to calculate how much fuel to put in to leave a 250mm air gap.

    Any help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by sharlen View Post
    This will probably be quite simple to work out, but my maths A Level was 14 years ago, and it's all a bit vague now.

    We have a petrol tank (a cylinder) which is 5280mm long and orientated horizontally (i.e. on its side). The volume of this tank is 15,000 Ltrs.

    We need to fill this tank, BUT leaving a 250mm air gap at the top.

    Obviously if the tank were orientated vertically it makes life easier (15,000 divided by 5280 = the litres per millimetre x 250mm = the litres occupying a 250mm gap. Deduct this from the 15,000 = the amount of fuel to put into the tank leaving a 250mm gap.).

    The tank is however on its side with the filling cap centrally located. We therefore have the incremental curvature of the tank to consider and we are at a loss to how to calculate how much fuel to put in to leave a 250mm air gap.

    Any help would be greatly appreciated.
    Put in 15,000 - 1164.6 = 13835.4 litres.

    I'll post details of the calculation a bit later.
    Last edited by mr fantastic; March 6th 2008 at 12:01 AM.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Put in 15,000 - 1164.6 = 13835.4 litres.

    I'll post details of the calculation a bit later.
    15,000 litres = 15 m^3.

    Volume of cylinder = pi r^2 h.

    Therefore r = 0.951 m.

    Volume of 'gap' = (area of circle segment) (length of cylinder).

    Area of circle segment = 0.5 (0.951)^2 (1.484 - sin(1.484)) = 0.220567 m^2.

    The angle 1.484 radians is calculated from 2 \times \cos^{-1} \frac{0.701}{0.951}, where the 0.701 comes from 0.951 - 0.250.

    Therefore volume of gap = (0.220567) (5.280) = 1.164595 m^3 = 1164.6 litres.
    Last edited by mr fantastic; March 6th 2008 at 12:18 AM.
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