Homework Help: Related rates weight of duckling

1. May 5, 2013

syeh

1. The problem statement, all variables and given/known data
Suppose that during the first year after its hatching, the weight of a duck increases at a rate proportional to its weight. The duck weighed 2 pounds when hatched, and 3.5 lbs at age 4 months. How many lbs will it weight at 6 months?

A) 4.2 lbs
B) 4.6 lbs
C) 4.8 lbs
D) 5.6 lbs
E) 6.5 lbs

2. Relevant equations

3. The attempt at a solution

I assumed this was a linear equation and used the points (0, 2) and (4, 3.5) to find the slope, .375, and the equation of the line to be y=.375x+2. Then i plugged in 6 and got y(6)=.375(6)+2 = 4.25. But the answer is 4.6.

Is the graph not a linear equation? Maybe since it says "weight increases at a rate proportional to its weight", it is an exponential function?? Then, how would I find f(6)??

2. May 5, 2013

rock.freak667

If dw/dt is the rate of change of weight, then

dw/dt ∝ w such that dw/dt = kw where k is a constant.

You will need to solve this DE to get w(t).

Then they gave you two conditions, so that you can solve for your constants.

3. May 5, 2013

syeh

What do you mean, dw/dt=kw? by using (0,2) and (4,3.5), how would i solve for w(t)??

4. May 5, 2013

SteamKing

Staff Emeritus
Read the problem statement carefully. "the weight of a duck increases at a rate proportional to its weight."

Thus, let w(t) = the weight of the duck at time t.

the rate of change in weight with respect to time is dw/dt.

This rate is proportional to the weight of the duck, thus dw/dt = kw, where k = constant of proportionality.

Do you know about separation of variables?

5. May 5, 2013

syeh

ok, I see. So i took dw/dt = kw and got
∫1/w dw = ∫k dt
lnw = kt + C

Using (0,2) to find C:
ln2 = C

Using (4, 3.5) to find k:
ln3.5 = 4.5k + ln2
4.5k = ln(1.75)
k=1.124

So, lnw= 0.124t + ln2
w=2e^(0.124t)

to plug in 6 months:
w(6)= 4.218

How come I didnt get the correct answer, 4.6 lbs?

6. May 5, 2013

haruspex

4 became 4.5?

7. May 5, 2013

Thank you