I've been trying to solve this problem for days without involving a stat program or anything similar (including least squares fitting) but I've been unsuccessful. I attempted to substitute the given points, form three equations but I couldn't solve them. How do I get around this problem? In advance thank you for your help.
Define $B:=Ae^{-14k},\,t:=e^{-5k}$ so $B+C=9176,\,Bt^{11}+C=7681,\,Bt^{13}+C=7542$ and $B(1-t^{11})=1495,\,Bt^{11}(1-t^2)=139$. Hence $1495t^{13}-1634t^{11}+139=0$. The only root in $(0,\,1)$ is $t=0.9082$. Now use$$k=-\frac15\ln t,\,\left(\begin{array}{c} B\\ C \end{array}\right)=\left(\begin{array}{cc} 1 & 1\\ t^{11} & 1 \end{array}\right)^{-1}\left(\begin{array}{c} 9176\\ 7681 \end{array}\right),\,A=Bt^{-2.8}.$$