(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A 5-gallon bucket is full of pure water. Suppose we begin dumping salt into the bucket at a rate of .25 pounds per minute. Also, we open the spigot so that .5 gallons per minute leaves the bucket, and we add water to keep the bucket full. If the saltwater solution is always well mixed, what is the amount of salt in the bucket after

a) 1 minute?

b) 10 minutes?

c) 60 minutes?

d) 1000 minutes?

e) a very, very long time?

2. Relevant equations

16 ounces = 1 pound

1 gallon = 128 oz

dSalt/dt = .25 lbs/min

dSolution/dt = -.5 gal/min

3. The attempt at a solution

At first, I just set this up like a simple ratio problem...

.25 pounds salt = 4 ounces

5 gallons water = 640 ounces

4 oz salt / 640 oz water = x oz salt / 576 oz water

x = (576*4)/(640) = 3.6 oz salt = .225 pounds salt

However, I checked in the back of the book and the answer listed is .238 pounds of salt. I don't understand how they came up with that answer?

Also, I'm a little confused as to how to incorporate the diff. eqs. into the solution. I don't think the way I solved it was right because I mostly just used alegbra. I think there should be a way to use both the rate that the salt is being poured in and the rate at which the solution is leaking out, but I don't know exactly how. If I can come up with the equation, I think I could do the rest of the integration and stuff, but it seems like I'm just stuck on the first part.

Any help would be greatly appreciated. Thanks in advance :)

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# Homework Help: First-Order Diff. Eq.

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