Step 1 :Given values: \(G = 6.674 \times 10^{-11} \mathrm{m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\), \(M_{\oplus} = 5.97 \times 10^{24} \mathrm{~kg}\), \(R_{\oplus} = 6.37 \times 10^{6} \mathrm{~m}\), and altitude = 420,000 m.
Step 2 :Calculate the distance r from the center of Earth to the ISS: \(r = R_{\oplus} + \text{altitude} = 6.37 \times 10^{6} \mathrm{~m} + 420,000 \mathrm{~m} = 6.79 \times 10^{6} \mathrm{~m}\)
Step 3 :Use the formula for orbital speed: \(v = \sqrt{\frac{G \times M_{\oplus}}{r}}\)
Step 4 :Plug in the values: \(v = \sqrt{\frac{6.674 \times 10^{-11} \mathrm{m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2} \times 5.97 \times 10^{24} \mathrm{~kg}}{6.79 \times 10^{6} \mathrm{~m}}}\)
Step 5 :Calculate the orbital speed: \(v \approx 7660 \mathrm{~m} / \mathrm{s}\)
Step 6 :\(\boxed{\text{The approximate orbital speed of the ISS is 7660 m/s (Option C)}}\)