Let S be the set of ordered triples (x, y, z) of real numbers for which
\log _{10}(x+y)=z \quad \text { and } \quad \log _{10}\left(x^{2}+y^{2}\right)=z+1
There are real numbers a and b such that for all ordered triples (x, y, z) in S we have x^{3}+y^{3}=a \cdot 10^{3 z}+b \cdot 10^{2 z}. What is the value of a+b ?
Answer Choices
A. \dfrac{15}{2}
B. \dfrac{29}{2}
C. 15
D. \dfrac{39}{2}
E. 24