Hey,
Here's a surface area ratio problem of which I could use some help:
Titan and Rhea are the two largest moons of Saturn. The ratio of the surface area of Titan to the surface area of Rhea is about 34:3. If the diameter of Rhea is about 1530 km, what is the diameter of Titan, to the nearest ten kilometres?
Merci!
Lengths scale as the square root of areas. So, as heureka has calculated,
$${\mathtt{radiusofRhea}} = {\sqrt{{\frac{{\mathtt{34}}}{{\mathtt{3}}}}}}{\mathtt{\,\times\,}}{\mathtt{1\,530}} \Rightarrow {\mathtt{radiusofRhea}} = {\mathtt{5\,150.747\: \!518\: \!564\: \!660\: \!29}}$$
Note that this is 5150km to the nearest ten kilometres.
$$\boxed{\mbox{Surface area of a sphere: } \quad S=\pi d^2}$$
$$S_{Titan}=\pi d_{Titan}^2$$
$$S_{Rhea}=\pi d_{Rhea}^2$$
$$\dfrac{S_{Titan}}
{S_{Rhea}}
=
\dfrac{\not{\pi} d_{Titan}^2}
{\not{\pi} d_{Rhea}^2}
=\dfrac{34}{3}$$
$$\dfrac{d_{Titan}^2}
{d_{Rhea}^2}
=\dfrac{34}{3}$$
$$d_{Titan}^2=d_{Rhea}^2*\left(\frac{34}{3} \right)$$
$$d_{Titan}=d_{Rhea}*\sqrt{\left(\frac{34}{3} \right)} \quad | \quad d_{Rhea}=1530\;km$$
$$\textstyle{d_{Titan}=1530\;km*\sqrt{\left(\frac{34}{3} \right)} =1530\;km*3.36650164612 = 5150.75\;km\approx5151\;km}$$
Diameter of Titan is 5151 km
Lengths scale as the square root of areas. So, as heureka has calculated,
$${\mathtt{radiusofRhea}} = {\sqrt{{\frac{{\mathtt{34}}}{{\mathtt{3}}}}}}{\mathtt{\,\times\,}}{\mathtt{1\,530}} \Rightarrow {\mathtt{radiusofRhea}} = {\mathtt{5\,150.747\: \!518\: \!564\: \!660\: \!29}}$$
Note that this is 5150km to the nearest ten kilometres.