Example $2$: Solve the preceding example using Newton$-$Raphson method

Solution:

$f_1=p-7\mathrm{.}2w^2-392\mathrm{.}4=0$

$f_2=p-810+25w_1+3\mathrm{.}75w_1^2=0$

$f_3=p-900+65w_2+30w_2^2=0$

$f_4=w-w_1-w_2=0$

Step $2$: Choose trial values of the variables, $p=750,w_1=3,w_2=1\mathrm{.}5,w=5$

Step $3$: calculate the magnitudes of the $f\text{'}$s at the temporary values of the variables: $f_1=177\mathrm{.}6,f_2=48\mathrm{.}75,f_3=15\mathrm{.}0,f_4=0\mathrm{.}50$

Complete the solution until the calculations converged satisfactorily. I need full steps and details