Ratio of volume of 2 spheres is 64 : 27 calculate the ratio of their surface area
Answers
Answer:
The ratio of the surface areas is 16 : 9
Step-by-step explanation:
The ratio of the surface areas is 16 : 9.
Step-by-step explanation:
To find the ratio of the surface areas, first we have to find the surface areas with their volumes.
Radius of big sphere = R
Radius of small sphere = r.
Volume of bigger sphere \bold{=\frac{4}{3} \pi R^{3}}=
3
4
πR
3
Volume of smaller sphere \bold{=\frac{4}{3} \pi r^{3}}=
3
4
πr
3
Given,
Volume of bigger sphere : Volume of smaller sphere = 64 : 27.
\begin{gathered}\begin{array}{l}{\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}=\frac{64}{27}} \\ {\Rightarrow > \frac{R^{3}}{r^{3}}=\frac{64}{27}}\end{array}\end{gathered}
3
4
πr
3
3
4
πR
3
=
27
64
⇒>
r
3
R
3
=
27
64
\begin{gathered}\begin{aligned}=& \sqrt[3]{\frac{R^{3}}{r^{3}}}=\sqrt[3]{\frac{64}{27}} \\ &= > \frac{R}{r}=\frac{4}{3} \end{aligned}\end{gathered}
=
3
r
3
R
3
=
3
27
64
=>
r
R
=
3
4
Surface area of bigger sphere =4 \pi R^{2}=4πR
2
Surface area of smaller sphere =4 \pi r^{2}=4πr
2
Hence, Surface area of bigger sphere: Surface area of smaller sphere = \begin{gathered}\begin{array}{l}\bold{{=4 \pi R^{2} : 4 \pi r^{2}}} \\ {=\frac{4 \pi R^{2}}{4 \pi r^{2}}=\frac{R^{2}}{r^{2}}} \\ {=(\frac{4}{3})^{2}=\frac{16}{9}}\end{array}\end{gathered}
=4πR
2
:4πr
2
=
4πr
2
4πR
2
=
r
2
R
2
=(
3
4
)
2
=
9
16
Thus, the ratio of their surface areas = 16 : 9