If the surface areas of two spheres are in the ratio of 9: 49, then the ratio of their volumes is:
27:343
To find the ratio of the volumes of two spheres given the ratio of their surface areas, we can use the formulas for surface area and volume of a sphere.
Let's denote the radii of the two spheres as r_1 and r_2.
The formula for the surface area of a sphere is given by:
A = 4\pi r^2
Therefore, the ratio of the surface areas of the two spheres is:
\frac{4\pi r_1^2}{4\pi r_2^2} = \frac{r_1^2}{r_2^2} = \frac{9}{49}
This implies:
\left(\frac{r_1}{r_2}\right)^2 = \frac{9}{49}
Taking the square root of both sides, we get:
\frac{r_1}{r_2} = \frac{3}{7}
Now, the formula for the volume of a sphere is:
V = \frac{4}{3}\pi r^3
So, the ratio of the volumes of the two spheres is:
\frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \frac{r_1^3}{r_2^3}
Substituting the known ratio of the radii, we have:
\frac{r_1^3}{r_2^3} = \left(\frac{3}{7}\right)^3 = \frac{27}{343}
Therefore, the ratio of the volumes of the two spheres is 27:343.
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