Sphere Formulas for Volume and Area

To solve this problem, we need the formulas for the volume and surface area of a sphere. Let the radius of the first sphere be $r_1$ and the radius of the second sphere be $r_2$.

• The formula for the volume ($V$) of a sphere is: $V = \frac{4}{3} \pi r^3$
• The formula for the surface area ($A$) of a sphere is: $A = 4 \pi r^2$

Calculating Radius Ratio from Volume Ratio

We are given that the ratio of the volumes of the two spheres is 27:8.

Let the volumes be $V_1$ and $V_2$. So, $$ \frac{V_1}{V_2} = \frac{27}{8} $$

Using the volume formula: $$ \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{r_1^3}{r_2^3} $$

Equating the two expressions for the volume ratio: $$ \frac{r_1^3}{r_2^3} = \frac{27}{8} $$

To find the ratio of the radii ($r_1:r_2$), we take the cube root of both sides: $$ \frac{r_1}{r_2} = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} $$

So, the ratio of the radii of the two spheres is 3:2.

Calculating Surface Area Ratio

Now, we need to find the ratio of their surface areas ($A_1:A_2$). Using the surface area formula: $$ \frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{r_1^2}{r_2^2} $$

This can be written as: $$ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 $$

We already found that the ratio of the radii $\frac{r_1}{r_2} = \frac{3}{2}$. Substituting this value: $$ \frac{A_1}{A_2} = \left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4} $$

Conclusion

Therefore, the ratio of the surface areas of the spheres is 9:4.