Multiple choice

A right circular cone and a sphere have equal volumes. If the cone has radius $x$ and height $2x$, calculate the radius of the sphere.

  1. $x$
  2. $\dfrac{x}{^3\sqrt{2}}$
  3. $^3\sqrt{2}$
  4. $\dfrac{1}{^3\sqrt{2}}$
  5. $^3\sqrt{2x}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The cone volume is (1/3)pi x^2(2x) = (2/3)pi x^3. Equating this to the sphere volume (4/3)pi r^3 gives r^3 = x^3/2. Hence, r = x/cuberoot(2).

AI explanation

Using the cone volume formula one third times pi times radius squared times height, the volume is one third times pi times x squared times 2x, which simplifies to two thirds times pi times x cubed. Setting this equal to the sphere volume formula four thirds times pi times radius cubed, we get two thirds times pi times x cubed equals four thirds times pi times radius cubed. Solving for the radius of the sphere gives radius cubed equals x cubed divided by 2, so the radius is x divided by the cube root of 2.

Keep practicing — related questions