Question

The volumes of two cones of equal height are in the ratio \( 1849 : 961 \). What is the ratio of their radii?

• A. \( \frac{41}{31} \)
• B. \( \frac{39}{31} \)
• C. \( \frac{43}{31} \)
• D. \( \frac{43}{29} \)

Hint:

When the heights of the cones are the same, the ratio of their volumes is the square of the ratio of their radii.

Answer 1

The Correct Option is C

The volume of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h, \] where \( r \) is the radius and \( h \) is the height. Since the heights of the cones are the same, the ratio of their volumes depends on the ratio of the squares of their radii: \[ \frac{V_1}{V_2} = \frac{r_1^2}{r_2^2}. \] Given that the ratio of the volumes is \( \frac{1849}{961} \), we have: \[ \frac{r_1^2}{r_2^2} = \frac{1849}{961} \quad \Rightarrow \quad \frac{r_1}{r_2} = \sqrt{\frac{1849}{961}} = \frac{43}{31}. \]