Certainly! To solve this problem, let's carefully follow these steps:
1. Understand Ratios and Volumes:
- The ratio of the radii of the first sphere to the second sphere is given as \( r_1 : r_2 = 2 : 3 \).
- We are given the volume of the second sphere, \( V_2 = 20 \, \text{cm}^3 \).
- We need to calculate the volume of the first sphere, \( V_1 \).
2. Volume Relationship in Spheres:
- The volume \( V \) of a sphere is related to its radius \( r \) by the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
- Therefore, the volume ratio of two spheres can be written in terms of the ratio of their radii:
[tex]\[
\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3
\][/tex]
3. Apply the Ratios Given:
- Substituting the given ratio of radii (\( \frac{r_1}{r_2} = \frac{2}{3} \)):
[tex]\[
\left(\frac{r_1}{r_2}\right)^3 = \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}
\][/tex]
4. Determine the Volume of the First Sphere:
- Using the ratio of the volumes:
[tex]\[
\frac{V_1}{V_2} = \frac{8}{27}
\][/tex]
- We know \( V_2 = 20 \, \text{cm}^3 \), so:
[tex]\[
V_1 = V_2 \times \frac{8}{27} = 20 \times \frac{8}{27}
\][/tex]
5. Calculate the Result:
- Perform the multiplication:
[tex]\[
V_1 = 20 \times \frac{8}{27} = \frac{160}{27}
\][/tex]
- Simplify the fraction:
[tex]\[
V_1 \approx 5.925925925925925
\][/tex]
So, the volume of the first sphere is approximately [tex]\( 5.93 \, \text{cm}^3 \)[/tex]. The precise value is [tex]\( 5.925925925925925 \, \text{cm}^3 \)[/tex].
