Answer
We note that the altitude of the rocket is $$h=R-R_{E}$$ where $$R_{E}=6.37 \times 10^{6} \mathrm{m} .$$ With $$M=5.98 \times 10^{24} \mathrm{kg}, R_{0}=R_{E}+h_{0}=6.57 \times 10^{6} \mathrm{m}$$ and $$R=7.37 \times 10^{6} \mathrm{m},$$ we have
$$
K_{i}+U_{i}=K+U \Rightarrow \frac{1}{2} m\left(3.70 \times 10^{3} \mathrm{m} / \mathrm{s}\right)^{2}-\frac{G m M}{R_{0}}=K-\frac{G m M}{R},
$$
which yields $$K=3.83 \times 10^{7} \mathrm{J} $$
Work Step by Step
We note that the altitude of the rocket is $$h=R-R_{E}$$ where $$R_{E}=6.37 \times 10^{6} \mathrm{m} .$$ With $$M=5.98 \times 10^{24} \mathrm{kg}, R_{0}=R_{E}+h_{0}=6.57 \times 10^{6} \mathrm{m}$$ and $$R=7.37 \times 10^{6} \mathrm{m},$$ we have
$$
K_{i}+U_{i}=K+U \Rightarrow \frac{1}{2} m\left(3.70 \times 10^{3} \mathrm{m} / \mathrm{s}\right)^{2}-\frac{G m M}{R_{0}}=K-\frac{G m M}{R},
$$
which yields $$K=3.83 \times 10^{7} \mathrm{J} $$
