69 points...XD
A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 
of air.If the internal diameter of dome is equal to its total height above the floor, find the height of the building.
Answers
Let r be the radius of hemisphere & Cylinder
and h be the height of the Cylinder, H be the
height of the Total building.
GIVEN :
Volume of air = 880/21 m³
Internal diameter (d) = H
Internal Diameter = 2r = H
Total Height of the building (H) = 2r……(1)
Height of the building = height of the cylinder +
radius of the hemispherical Dome
H = h + r
2r = h +r [from eq 1]
2r -r = h
r = h ……………..(2)
Volume of air inside the building = Volume of
cylindrical portion + Volume of hemispherical portion
πr²h + (2πr³/3)= 880/21
π(h)²h + (2π(h)³/3)= 880/21
[From eq 2, r= h]
πh³ + ⅔ πh³ = 880/21
πh³(1+⅔) = 880/21
πh³[(3+2)/3] = 880/21
πh³[5/3] = 880/21
22/7 × h³ × 5/3 = 880/21
h³ = (880 ×3 ×7) / 21 × 22 × 5
h³ = 40 /5 = 8
h³ = 8
h = ³√8 = ³√2×2×2
h = 2 m
h= r = 2 m [From eq 2, r= h]
Total height of the building( H) = 2r = 2×2 = 4 m
Hence, the Total height of the building is 4m.
Answer:
Let r be the radius of hemisphere & Cylinder
and h be the height of the Cylinder, H be the
height of the Total building.
GIVEN :
Volume of air = 880/21 m³
Internal diameter (d) = H
Internal Diameter = 2r = H
Total Height of the building (H) = 2r……(1)
Height of the building = height of the cylinder +
radius of the hemispherical Dome
H = h + r
2r = h +r [from eq 1]
2r -r = h
r = h ……………..(2)
Volume of air inside the building = Volume of
cylindrical portion + Volume of hemispherical portion
πr²h + (2πr³/3)= 880/21
π(h)²h + (2π(h)³/3)= 880/21
[From eq 2, r= h]
πh³ + ⅔ πh³ = 880/21
πh³(1+⅔) = 880/21
πh³[(3+2)/3] = 880/21
πh³[5/3] = 880/21
22/7 × h³ × 5/3 = 880/21
h³ = (880 ×3 ×7) / 21 × 22 × 5
h³ = 40 /5 = 8
h³ = 8
h = ³√8 = ³√2×2×2
h = 2 m
h= r = 2 m [From eq 2, r= h]
Total height of the building( H) = 2r = 2×2 = 4 m
Hence, the Total height of the building is 4m.