Find the area between an exponential tangent and normal

The Teacher supplies $dy/dx=e^x$, so the tangent slope at $Q$ is $e^t$. The tangent is $y-e^t=e^t(x-t)$. Setting $y=0$ gives $x=t-1$, hence $T(t-1,0)$.

The normal slope is $-1/e^t$, and its equation is $y-e^t=-e^{-t}(x-t)$. At $y=0$, $e^{2t}=x-t$, giving $N(t+e^{2t},0)$.

The base length is $NT=(t+e^{2t})-(t-1)=e^{2t}+1>0$. The height above the x-axis is $e^t>0$. Thus the area is $\tfrac12e^t(e^{2t}+1)$ square units, as printed in the Teacher solution. The Student copy has the labelled diagram but blank working. No restriction on real $t$ is stated; the drawing’s placement of $T$ to the right of the origin is illustrative, not a condition $t>1$.