How to calculate the values of \$R\$ and \$C\$?
Partial answer so far ...

Given \$V_C(\infty)=76.73V\$ and \$PR_2(\infty)=588.8mW\$ you could have immediately calculated R2: from \$ P = \frac {V^2}{R} \$ we can calculate

$$ R_2 = \frac {V^2}{P} = \frac {76.73^2}{0.5888} = 9999.14 ~ \Omega$$

That's close enough to your 10k.



Assuming that \$V(t)=94u_1(t)V\$ means the voltage steps from 0 to 94 V at t = 0 when C is completely discharged and behaving for an instant like a short circuit then we can use the other bits of information, \$iR_1(0^+)=20mA\$ and \$V(t)=94u_1(t)V\$, to calculate

$$ R_1 = \frac {V}{I} = \frac {94}{0.02} = 4700 ~ \Omega $$



If you can clarify what \$ u_1 \$ and \$ \delta \$ are I might be able to finish this.



Update: While I'm not familiar with Heaviside or delta dirac functions it seems that we don't have enough information to calculate the value of C. We need another voltage or current measurement at t > 0.