A building is in the form of a cylinder surmo | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the radius of the hemispherical dome be $r$ meters and the total height of the building be $h$ meters.Since the base diameter of the dome is e

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(D) Let the radius of the hemispherical dome be $r$ meters and the total height of the building be $h$ meters.Since the base diameter of the dome is equal to $\frac{2}{3}$ of the total height,therefore $2r = \frac{2}{3}h$,which implies $r = \frac{h}{3}$.Let $H$ be the height of the cylindrical portion. Then $H = h - r = h - \frac{h}{3} = \frac{2}{3}h$ meters.The volume of air inside the building is the sum of the volume of the hemispherical dome and the volume of the cylindrical portion.Volume $= \frac{2}{3}\pi r^3 + \pi r^2 H = \frac{2}{3}\pi (\frac{h}{3})^3 + \pi (\frac{h}{3})^2 (\frac{2}{3}h) = \frac{2}{3}\pi (\frac{h^3}{27}) + \pi (\frac{h^2}{9}) (\frac{2}{3}h) = \frac{2\pi h^3}{81} + \frac{2\pi h^3}{27} = \frac{2\pi h^3 + 6\pi h^3}{81} = \frac{8\pi h^3}{81}$.Given the volume is $67 \frac{1}{21} = \frac{1408}{21} \, m^3$.So,$\frac{8}{81} 	imes \frac{22}{7} 	imes h^3 = \frac{1408}{21}$.$h^3 = \frac{1408}{21} 	imes \frac{81 	imes 7}{8 	imes 22} = \frac{1408}{21} 	imes \frac{567}{176} = 8 	imes 27 = 216$.$h = \sqrt[3]{216} = 6 \, m$.

$A$ building is in the form of a cylinder surmounted by a hemispherical dome (see figure). The base diameter of the dome is equal to $\frac{2}{3}$ of the total height of the building. Find the height of the building,if it contains $67 \frac{1}{21} \, m^3$ of air. (in $m$)

Similar Questions

The barrel of a fountain pen,cylindrical in shape,is $7 \, cm$ long and $5 \, mm$ in diameter. $A$ full barrel of ink in the pen is used up on writing $3300$ words on an average. How many words can be written in a bottle of ink containing one-fifth of a litre?

How many balls of radius $0.5\, cm$ can be produced by melting a metallic cylinder with radius $4\, cm$ and height $3\, cm$?

$A$ well with radius $2 \, m$ and depth $50 \, m$ is dug and the soil taken out is spread evenly on a circular plot with radius $40 \, m$. Find the increase in the level of the plot in $cm$.

How many spherical lead shots,each of diameter $4.2 \, cm$,can be obtained from a solid rectangular lead piece with dimensions $66 \, cm \times 42 \, cm \times 21 \, cm$?

$A$ cylinder is closed at both the ends by $12 \, cm$ high cones. The radius of the cylinder is $5 \, cm$ and the total height of the combined article is $41 \, cm$. Find the total surface area of this article in $cm^2$. $(\pi = 3.14)$

Vedclass Test Series

Exam Paper Generator

Online Exam Module