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  • 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?

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41\frac{19}{21}m^3 of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building?

Answers (1)

Give: Volume of building

41\frac{19}{21}=\frac{880}{21}

            Let total height above the floor = h

            Hemisphere‘s diameter = h (given)

          Radius =\frac{h}{2}

          Volume=\frac{2}{3}\pi r^3=\frac{2}{3}\pi \left ( \frac{h}{2} \right )^3

            Height of cylinder = total height – the height of hemisphere

            h-\frac{h}{2}=\frac{h}{2}

            Volume \pi r^2 h=\pi \times \left (\frac{h}{2} \right )^2 \times \frac{h}{2} =\pi \left (\frac{h}{2} \right )^3

            According to question

            The volume of building = Volume of cylinder + volume of the hemisphere

                        \frac{880}{21}=\left ( \frac{h}{2} \right )^3\left [ \pi + \frac{2}{3} \pi \right ]

                        \frac{880}{21}=\left ( \frac{h}{2} \right )^3\left [ \frac{3\pi +2\pi }{3} \right ]

                        \frac{880 \times 2^3}{21}=h^3 \left [ \frac{5\pi }{3} \right ]

                        \frac{880 \times 2\times 2\times 2\times 7\times 3}{21\times 5\times 22}=h^3

                        h^3 = {2 \times 2 \times 2\times 2\times 2\times 2

                        h = \sqrt[3]{2 \times 2 \times 2\times 2\times 2\times 2}

                        h = 2 \times 2

                        h=4m

Posted by

infoexpert21

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