What is the height of the cone?

Consider the following for the next two (02) items that follow : A right conical cap just covers two spheres placed one above the other on a table such that it touches both the spheres. Let r be the radius of the smaller sphere and R be the radius of the bigger sphere. Let 2\theta be the vertical angle of the cone.

  1. A. \frac{2r^{2}}{R-r}
  2. B. \frac{2R^{2}}{R-r} ✓
  3. C. \frac{2(r^{2}+R^{2})}{R-r}
  4. D. \frac{r^{2}+R^{2}}{R-r}

Correct Answer: B. \frac{2R^{2}}{R-r}

Explanation

Let the cone's vertex distance to the large sphere's center be x. Since \sin \theta = \frac{R}{x} and the distance ratio gives \sin \theta = \frac{R-r}{R+r}, we find x = \frac{R(R+r)}{R-r}. The cone's total height is from vertex to table, x + R = R\left(\frac{R+r}{R-r}\right) + R = \frac{2R^2}{R-r}.

Related questions on Mensuration

Practice more CDS Elementary Mathematics questions