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
