A building is in the form of cylinder surmounted by a hemisphere vaulted dome and contains 1144/21?
... metre cube of air. If internal diameter of dome is equal to 4/5 of total height above the floor find the height of the building or:... metre cub
... metre cube of air. If internal diameter of dome is equal to 4/5 of total height above the floor find the height of the building
or:... metre cube of air. If internal diameter of dome is equal to 4/5 of total height above the floor find the height of the building
or:Volume of a cylinder c = l\u03c0r^2Volume of a hemisphere s = (1/2)(4/3)\u03c0r^3Given: l = h/5r = 4h/5 c + s = (\u03c0r^2)/5 + 4((1/2)(4/3)\u03c0r^3)/5 = 1144/21 The rule is you can do any valid operation on both sides of an equation and it will still be equal. Multiply by 21.21(\u03c0r^2)/5 + 84((1/2)(4/3)\u03c0r^3)/5 = 1144 Multiply by 5.21(\u03c0r^2) + 84((1/2)(4/3)\u03c0r^3) = 5720 Multiply by 6.126\u03c0r^2 + 336\u03c0r^3 = 34320Now you have a simple cubic which you may evaluate any way you know how. www.wolframalpha.com/input/?i=solve+126%CF%80r%5E2+%2B+336%CF%80r%5E3+%3D+34320
or:h = internal height of the buildingThe diameter d of the building is:d = 4/5 * hThe radius r is:r = 1/2 * d = (1/2)(4/5) h = 2/5 * h As the dome is a hemisphere, the (internal) height of this hemisphere is also 2/5 * h. It follows that the height of the cylindrical part of the building is 3/5 * h.The volume of our specific cylinder c is then:c = \u03c0 (2/5 * h)^2 * 3/5 * h = \u03c0 4/25 * h^2 * 3/5 * h = \u03c0 12/125 * h^3The volume of the hemisphere s is:s = \u03c0 (1/2)(4/3) (2/5 * h)^3 = \u03c0 (4/6)(8/125) * h^3The total volume v = c + s equals 1144/21. That takes a few steps of simplifying and finally isolating h:1144 / 21 = 12/125 * \u03c0 h^3 + (4/6)(8/125) \u03c0 h^34 * 286 / 21 = 4 \u03c0 h^3 (3/125 + (1/6)(8/125)) 286 / 21 = \u03c0 h^3 (3/125 + 4/375)286 / 21 = \u03c0 h^3 * 13/375107250 / 273 = \u03c0 h^3392.857142 / \u03c0 = h^3 h = 125.0503124^(1/3)h= 5.000670742The height of the building is 5.000670742 meters.
