If a cone and sphere have equal radii and volumes, then determine the ratio of the diameter of the sphere to the height of the cone?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1:2
The volume of a cone is \(\frac{1}{3}\pi r^2h\) and that of a sphere is \(\frac{4}{3}\pi r^3\). With the same radius \(r\), equating the two volumes gives the height of the cone in terms of \(r\).
Equal volumes (the same radius \(r\)):
\(\displaystyle \frac{1}{3}\pi r^2h=\frac{4}{3}\pi r^3\)
\(h=4r\)
The ratio (diameter of the sphere to height of the cone):
Diameter of the sphere \(=2r\)
Diameter of the sphere : height of the cone \(=2r:4r=1:2\)