How to solve this problem step by step with application of integration?

enter image source here

1 Answer
Mar 4, 2018

#a)# #N(14)=3100-400sqrt2~~2534#

#color(white)(...|)N(34)=3900-400sqrt2~~3334#

#b)# #N(t)=400sqrt(t+2)+1500-400sqrt2#

Explanation:

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#