A building is a cylinder surmounted by a hemispherical dome and contains \(41\dfrac{19}{21}\,\text{m}^3\) of air. If the internal diameter of the dome equals the total height above the floor, find the height of the building.
4 m
Step 1: Convert the given volume into an improper fraction.
\(41\dfrac{19}{21} = \dfrac{41 \times 21 + 19}{21} = \dfrac{861 + 19}{21} = \dfrac{880}{21}\,\text{m}^3.\)
Step 2: Let the radius of the hemispherical dome be \(r\,\text{m}.\)
The internal diameter of the dome = total height of the building.
So, \(2r = h + r \;\Rightarrow\; h = r.\)
This means the cylindrical part and the radius of the dome are equal in height.
Step 3: Write the formula for total volume of the building.
Total volume = Volume of cylinder + Volume of hemisphere
\(= \pi r^2h + \dfrac{2}{3}\pi r^3\)
Step 4: Substitute \(h = r\).
Volume \(= \pi r^2(r) + \dfrac{2}{3}\pi r^3\)
\(= \pi r^3 + \dfrac{2}{3}\pi r^3\)
\(= \dfrac{5}{3}\pi r^3.\)
Step 5: Equate with given volume.
\(\dfrac{5}{3}\pi r^3 = \dfrac{880}{21}\)
Step 6: Take \(\pi = \dfrac{22}{7}.\)
\(\dfrac{5}{3} \times \dfrac{22}{7} r^3 = \dfrac{880}{21}\)
\(\dfrac{110}{21} r^3 = \dfrac{880}{21}\)
Step 7: Simplify.
\(110 r^3 = 880\)
\(r^3 = 8\)
\(r = 2\,\text{m}.\)
Step 8: Find the height of the cylindrical part.
Since \(h = r = 2\,\text{m},\) the cylinder’s height = 2 m.
Step 9: Find the total height of the building.
Total height = Height of cylinder + Radius of dome
= \(2 + 2 = 4\,\text{m}.\)
Final Answer: The height of the building = 4 m.