documentclass[border=5mm]{standalone}
usepackage{tikz}

usetikzlibrary{calc,intersections}

[![enter image description here][1]][1]
begin{document}
begin{tikzpicture}[xscale=0.015,yscale=0.01]

draw[domain=0:500,samples=1000,variable=x,name path=A] plot ({x},{475.-712.*exp(-0.133e-2*x*10)+238.*exp(-0.400e-2*x*10)});
draw[-latex] (0,0)coordinate(O)-- (500,0) node[above]{$t$};
draw[-latex] (O) -- (0,500) node[right]{$theta_s(t)$};

draw[dashed,thick,name path=S100] (0,{475}) --  (500,{475});

draw[dashed,thick,name path=S95] (0,{475*0.95}) --  (500,{475*0.95});
path[name intersections={of=S95 and A,by=E}];


draw (E) |- (O-|E);

end{tikzpicture}
end{document}