Two metallic right circular cones having the same base radius 2.1 cm and heights 4.1 cm and 4.3 cm respectively are melted together and recast into a sphere. Find the radius of the sphere.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 2.1 cm
When solids are melted and recast, the volume does not change, so the volume of the sphere equals the total volume of the two cones. The cones have the same base, so together they hold as much as one cone of height 4.1 + 4.3 = 8.4 cm.
Volume of the two cones \(\displaystyle=\frac{1}{3}\pi r^2(h_1+h_2)=\frac{1}{3}\pi(2.1)^2(8.4)\)
Volume of the sphere \(\displaystyle=\frac{4}{3}\pi R^3\)
\(\displaystyle\frac{4}{3}\pi R^3=\frac{1}{3}\pi(2.1)^2(8.4)\), so \(4R^3=(2.1)^2\times8.4=4\times(2.1)^3\), since \(8.4=4\times2.1\)
\(R^3=(2.1)^3\), so \(R=2.1\,\mathrm{cm}\)