... 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
Answers (2)
Volume of a cylinder c = lπr^2
Volume of a hemisphere s = (1/2)(4/3)πr^3
Given:
l = h/5
r = 4h/5
c + s = (πr^2)/5 + 4((1/2)(4/3)πr^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(πr^2)/5 + 84((1/2)(4/3)πr^3)/5 = 1144 Multiply by 5.
21(πr^2) + 84((1/2)(4/3)πr^3) = 5720 Multiply by 6.
126πr^2 + 336πr^3 = 34320
Now 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
h = internal height of the building
The diameter d of the building is:
d = 4/5 * h
The 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 = π (2/5 * h)^2 * 3/5 * h = π 4/25 * h^2 * 3/5 * h = π 12/125 * h^3
The volume of the hemisphere s is:
s = π (1/2)(4/3) (2/5 * h)^3 = π (4/6)(8/125) * h^3
The total volume v = c + s equals 1144/21. That takes a few steps of simplifying and finally isolating h:
1144 / 21 = 12/125 * π h^3 + (4/6)(8/125) π h^3
4 * 286 / 21 = 4 π h^3 (3/125 + (1/6)(8/125))
286 / 21 = π h^3 (3/125 + 4/375)
286 / 21 = π h^3 * 13/375
107250 / 273 = π h^3
392.857142 / π = h^3
h = 125.0503124^(1/3)
h= 5.000670742
The height of the building is 5.000670742 meters.