1. First, convert the model distance to base SI.
[tex]30\,\mathrm{cm} = 3\times10^{-1}\,\mathrm m[/tex]
Then the ratio of model distance to actual distance is
[tex]\dfrac{30\,\rm cm}{1.50\times10^{11}\,\rm m} = \dfrac{3\times10^{-1}\,\mathrm m}{1.50\times10^{11}\,\rm m} = \dfrac{2}{10^{12}} = \dfrac{2}{2\times5\times10^{11}} = \dfrac{1}{5\times10^{11}}[/tex]
or 1 : 5 × 10¹¹.
2. The model/actual distance ratio should be the same for each planet.
Mercury:
[tex]12.6\,\mathrm{cm} = 1.26\times10^{-1}\,\mathrm m[/tex]
[tex]\dfrac{12.6\,\rm cm}{5.80\times10^{10}\,\rm m} = \dfrac{1.26\times10^{-1}\,\rm m}{5.80\times10^{10}\,\rm m} \approx \dfrac{1}{4.6\times10^{11}}[/tex]
or 1 : 4.6 × 10¹¹.
Venus:
[tex]21.6\,\mathrm{cm} = 2.16\times10^{-1}\,\rm m[/tex]
[tex]\dfrac{21.6\,\rm cm}{1.08\times10^{11}\,\rm m} = \dfrac{2.16\times10^{-1}\,\rm m}{1.08\times10^{11}\,\rm m} = \dfrac{2}{10^{12}} = \dfrac{2}{2\times5\times10^{11}} = \dfrac1{5\times10^{11}}[/tex]
or 1 : 5 × 10¹¹.
Mercury's distance is the incorrect one. The correct model distance should be [tex]x[/tex] such that
[tex]\dfrac{x\,\rm cm}{5.80\times10^{10}\,\rm m} = \dfrac{1}{5\times10^{11}}[/tex]
Solve for [tex]x[/tex].
[tex]x\,\mathrm{cm} = \dfrac{5.80\times10^{10}\,\rm m}{5\times10^{11}}[/tex]
[tex]x\,\mathrm{cm} = \dfrac{1.16\,\rm m}{10^1}[/tex]
[tex]x\,\mathrm{cm} = 1.16\times10^{-1}\,\rm m[/tex]
[tex]x\,\mathrm{cm} = \boxed{11.6\,\rm cm}[/tex]
3.The actual distance for Mars is [tex]y[/tex] such that
[tex]\dfrac{34.4\,\rm cm}{y\,\rm m} = \dfrac{1}{5\times10^{11}}[/tex]
Solve for [tex]y[/tex].
[tex]34.4\times5\times10^{11} \,\mathrm{cm} = y\,\rm m[/tex]
[tex]y\,\mathrm m = 172 \times10^{11}\,\rm cm[/tex]
[tex]y\,\mathrm m = 172 \times10^{13}\,\rm m[/tex]
[tex]y\,\mathrm m = \boxed{1.72} \times10^{\boxed{11}}\,\rm m[/tex]
4. Divide the distance by the speed of light to recover the time.
[tex]\dfrac{7.92\times10^{11}\,\rm m}{3.0\times10^8\frac{\rm m}{\rm s}} = 2.64\times10^3\,\mathrm s[/tex]
Convert to minutes.
[tex]2.64\times10^3\,\mathrm s \cdot \dfrac{1\,\rm min}{60\,\rm s} = 0.044\times10^3\,\mathrm{min} = \boxed{44}\,\rm min[/tex]
