To find the ratio of the radii of two spheres given that their volumes are in a ratio of 18, we can use the formula for the volume of a sphere:
Volume of a sphere: V = \frac{4}{3} \pi r^3
Let the volumes of the two spheres be V_1 and V_2. According to the problem:
\frac{V_1}{V_2} = 18
Now, substituting the volume formula into the ratio gives:
\frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = 18
The \frac{4}{3} \pi in both the numerator and denominator cancels out, simplifying our equation to:
\frac{r_1^3}{r_2^3} = 18
To find the ratio of the radii, we take the cube root of both sides:
\frac{r_1}{r_2} = \sqrt[3]{18}
This means that the ratio of the radii of the two spheres is:
\frac{r_1}{r_2} = 18^{1/3}
The cube root of 18 can be approximated as:
18^{1/3} \approx 2.6207
Therefore, the ratio of the radii simplifies to:
Ratio of the radii: r_1 : r_2 \approx 2.62 : 1
In conclusion, if the volumes of two spheres are in a ratio of 18, the ratio of their radii is approximately 2.62:1.