This is a normal distribution.

We can say that the chances that a nail is shorter than 41.6mm is 2.5%

And the chances that a nail is longer than 49.7mm is 16%

$\displaystyle P(X < 41.6) = 0.025$

$\displaystyle P \left( \frac{X - \mu}{\sigma} < \frac{41.6 - \mu}{\sigma} \right) = 0.025$

$\displaystyle P \left( Z < \frac{41.6 - \mu}{\sigma} \right) = 0.025$

$\displaystyle \Phi \left( \frac{41.6 - \mu}{\sigma} \right) = 0.025$

$\displaystyle \frac{41.6 - \mu}{\sigma} = -1.96$

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$\displaystyle P(X > 49.7) = 0.16$

$\displaystyle 1 - P(X \leq 49.7) = 0.16$

$\displaystyle P \left( \frac{X - \mu}{\sigma} \leq \frac{49.7 - \mu}{\sigma} \right) = $

$\displaystyle P \left( Z \leq \frac{49.7 - \mu}{\sigma} \right) = 0.84$

$\displaystyle \Phi \left( \frac{49.7 - \mu}{\sigma} \right) = 0.84$

$\displaystyle \frac{49.7 - \mu}{\sigma} = 0.996$

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From here on it's simply a matter of setting those two equations equal to each other and solving.