#### Answer

The maximum value of $\Delta S$ is $0.034~S$

#### Work Step by Step

$\Delta S = 0.034~S~sin~[\frac{2\pi~(82.5-N)}{365.25}]$
The maximum value of $\Delta S$ occurs when $sin~[\frac{2\pi~(82.5-N)}{365.25}] = 1$
Then $\Delta S = 0.034~S$
The maximum value of $\Delta S$ is $0.034~S$