$8$ For a European call option on a nondividend paying stock with one year to expiry:

$($i$)$ The stock follows the Black$-$Scholes framework.

$($ii$)$ The price of the stock is $40.$

$($iii$)$ The strike price is $50.$

$($iv$)$ Gamma for the option is $0\mathrm{.}0425.$

$($v$)$ The continuously compounded risk$-$free interest rate is $0\mathrm{.}07.$

The price of the stock jumps to $40\mathrm{.}50$ and the price of the option increases by $0\mathrm{.}0915.$ Determine the implied volatility of the stock based on the delta$-$gamma approximation.