PLEASE HELP ME WITH THIS CALCULUS QUESTION!!! EASY 10 POINTS!!!

A puppy weighs 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weight when it is 3 months old?

a) 4.2 pounds

b) 4.6 pounds

c) 4.8 pounds

d) 5.6 pounds

e) 6.5 pounds

the answer is B, i just wanna know how to get it. please explain each step, thankyou!!!! =D

The problem is a straight forward calculation of a separable differential equation. Since the problem states that the rate of change of weight is directly proportional to its weight, then we have the linear ODE as
dw/dt = k w
Notice the right side has the direct proportional (the constant k) and the weight w. Using the separation and integrating both sides we obtain
int( dw/w, w) = int(k dt, t)
Integrating both sides we obtain
ln|w| = k t + C
expatiating both sides, and using properties of exponents we obtain
w(t) = exp(kt+C) = exp(kt)exp(C)= A exp(kt)
where A=exp(C). Notice we have two constants we need to figure out, so we may use this data now.
w(0) = 2.0 = A exp(0) = A
so we found that A = 2.0. Now we need to find k, so we know that when t = 2 and w = 3.5 and we have
w(2) = 3.5 = 2.0 exp(2k)
solving for k we obtain
k = 0.5 ln(3.5/2.0) ~ 0.2798
Now to find the weight at t = 3 months we have
w(3) = 2.0 exp(0.2798 * 3) ~ 4.63 ~ 4.6
Hope that helps

Well...if the rate of increase is proportional to it's weight, then I guess you would first have to do this.

Rate of increase refers to gradient. So what you would do is treat the weight of the puppy as a function.

So in this case; let the function of the puppy's weight be y.

So dy/dt (time being in months) = ky (where y is the weight of the puppy).

So dy/y = kdt. (then integrate both sides)

Then ln y = kt + c