Write True or False and justify your answer: The volume of a sphere is equal to two-thirds of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.
(A) Let the radius of the sphere be $r$.
Given that the height and diameter of the cylinder are equal to the diameter of the sphere.
Since the diameter of the sphere is $2r$,the height of the cylinder $h = 2r$ and the diameter of the cylinder is $2r$.
Therefore,the radius of the cylinder is $r$.
Volume of the sphere $= \frac{4}{3} \pi r^3$.
Volume of the cylinder $= \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3$.
Two-thirds of the volume of the cylinder $= \frac{2}{3} \times (2 \pi r^3) = \frac{4}{3} \pi r^3$.
Since the volume of the sphere is equal to $\frac{4}{3} \pi r^3$,the statement is True.
Given that the height and diameter of the cylinder are equal to the diameter of the sphere.
Since the diameter of the sphere is $2r$,the height of the cylinder $h = 2r$ and the diameter of the cylinder is $2r$.
Therefore,the radius of the cylinder is $r$.
Volume of the sphere $= \frac{4}{3} \pi r^3$.
Volume of the cylinder $= \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3$.
Two-thirds of the volume of the cylinder $= \frac{2}{3} \times (2 \pi r^3) = \frac{4}{3} \pi r^3$.
Since the volume of the sphere is equal to $\frac{4}{3} \pi r^3$,the statement is True.
Explore More
Similar Questions
State whether the following statement is true or false:
An open box measuring $12\, cm \times 4\, cm \times 3\, cm$ can hold a bar of $15\, cm$ in length.
An open box measuring $12\, cm \times 4\, cm \times 3\, cm$ can hold a bar of $15\, cm$ in length.
Easy
View Solution$A$ sphere and a cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height?
Medium
View SolutionFind the curved surface area of a cylinder whose height is $35 \, cm$ and radius of the base is $15 \, cm$. (in $cm^2$)
Easy
View SolutionThe surface area of a sphere is $5544 \, cm^2$. Find its volume in $cm^3$.
Medium
View SolutionThe length of a metal hollow cylinder is $70\, cm$. If the radius of the hollow inside it is $10\, cm$ and the metal thickness is $1\, cm$,find the difference between the areas of the outer and inner curved surfaces.
Medium
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo