We begin by solving for #N(t)#. We can do this by simply integrating both sides of the equation:
#N'(t)=200(t+2)^(-1/2)#
#int\ N'(t)\ dt=int\ 200(t+2)^(-1/2)\ dt#
We could do a u-substitution with #u=t+2# to evaluate the integral, but we recognize that #du=dt#, so we can just pretend #t+2# is a variable and use the power rule:
#N(t)=(200(t+2)^(1/2))/(1/2)+C=400sqrt(t+2)+C#
We can solve for the constant #C# since we know that #N(0)=1500#:
#N(0)=400sqrt(0+2)+C=1500#
#C=1500-400sqrt2#
This gives that our function, #N(t)# can be expressed as:
#N(t)=400sqrt(t+2)+1500-400sqrt2#
We can then plug in #14# and #34# to get the answers to part #A#:
#N(14)=400sqrt(14+2)+1500-400sqrt2=3100-400sqrt2~~2534#
#N(34)=400sqrt(34+2)+1500-400sqrt2=3900-400sqrt2~~3334#
