= Solution
The <finite-horizon GWP lifetime sensitivity> follows because a longer lifetime leaves more <methane> from a given pulse at later times and raises the steady burden for a given constant emission rate. For the stipulated exponential pulse response and unchanged <greenhouse gas radiative efficiency>, the integrated <methane> forcing over horizon $H$ is proportional to $\tau(1-e^{-H/\tau})$. The same <carbon dioxide> reference pulse appears in both <global warming potential> denominators, so it cancels from their ratio:
$$
\frac{\mathrm{GWP}_{20,1}}{\mathrm{GWP}_{20,0}}
=\frac{9.8492(1-e^{-20/9.8492})}{8.2540(1-e^{-20/8.2540})}
=1.13748.
$$
Thus \b[the twenty-year GWP increases by $13.75\%$] under these fixed-efficiency assumptions. This is smaller than the $19.33\%$ lifetime increase because part of the extra forcing occurs beyond the finite horizon. For an infinite horizon the fractional increase would approach the lifetime ratio. Indirect forcing and the distinction between a chemical lifetime and a <hydroxyl radical>-coupled <methane perturbation lifetime> require a fuller chemical model; they are held fixed in this comparison.
\Image[/past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-id-methane-lifetime.png]
{title=Original exponential methane-pulse responses and finite-horizon GWP sensitivity}
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