The volume of a cone and a sphere are equal. If the radius of the sphere is 50\% of the radius of the cone, then find the ratio of the radius of the sphere to the height of the cone.

Question

Accepted Answer

**Solution:**

1. Let the radius of the cone be *r* and the height of the cone be *h*.

2. Given, the radius of the sphere is 50\% of the radius of the cone, so radius of sphere = $ $\frac{r}{2}$ $.

3. Volume of the cone is given by:
   $$ V_{cone} = \frac{1}{3} \pi r^2 h $$

4. Volume of the sphere is given by:
   $$ V_{sphere} = \frac{4}{3} \pi \left( \frac{r}{2} \right)^3 = \frac{4}{3} \pi \frac{r^3}{8} = \frac{\pi r^3}{6} $$

5. Since volumes are equal:
   $$ \frac{1}{3} \pi r^2 h = \frac{\pi r^3}{6} $$

6. Simplify the equation:
   $$ \frac{1}{3} r^2 h = \frac{r^3}{6} $$

7. Multiply both sides by 6:
   $$ 2 r^2 h = r^3 $$

8. Divide both sides by $ r^2 $ (assuming $ r $\neq $0 $):
   $$ 2 h = r $$

9. Therefore:
   $$ h = \frac{r}{2} $$

10. The radius of the sphere is $ $\frac{r}{2}$ $, and the height of the cone is $$ \frac{r}{2}$ $.

11. Hence, the ratio of the radius of the sphere to the height of the cone is:
    $$ \frac{\frac{r}{2}}{\frac{r}{2}} = 1 : 1 $$

**Answer:** 1 : 1

Answer

Answer

Answer

Answer