Water Mass in Leaking Container

[IsoGD]

Homework Statement

Water is poured into a container that has a leak. The mass m of the water is given as a function of time t by m = 6.00t ^0.8 - 2.75t + 22.00, with t

0, m in grams, and t in seconds.

(a) At what time is the water mass greatest? (s)
(b) What is that greatest mass? (g)
(c) In kilograms per minute, what is the rate of mass change at t = 2.00 s? (kg/min)
(d) In kilograms per minute, what is the rate of mass change at t = 5.00 s? (kg/min)

Homework Equations

m = 6.00t ^0.8 - 2.75t + 22.00

The Attempt at a Solution

I didn't really have any clue on how to start this problem at all, so I was just able to find that the greatest mass is ~33.1 grams from guessing and checking. After that, I tried to see if I could work out what the time was by plugging the mass back into the equation, but the answer I got (17.7 seconds) was showing as incorrect on Webassign. So I really just need to be explained how to even start this problem in the first place so that I can work from there.

*Edited superscripts in*

 

[haruspex]

6.00t 0.8 - 2.75t + 22.00, with t

0, m in grams, and t in seconds.

From your answer, I deduce this is supposed to be 6.00t^0.8. You can use the X² button above the typing area to get superscripts.
How do you normally find maxima and minima of functions? Haven't you been taught a calculus method?

 

[IsoGD]

From your answer, I deduce this is supposed to be 6.00t^0.8. You can use the X² button above the typing area to get superscripts.
How do you normally find maxima and minima of functions? Haven't you been taught a calculus method?

Sorry about that, didn't notice the mistake. And this homework was assigned on the first day, Friday the 18th, without any prior instruction, and I'm taking Calculus and this Physics class at the same time so I haven't learned any methods for this.

 

[haruspex]

Sorry about that, didn't notice the mistake. And this homework was assigned on the first day, Friday the 18th, without any prior instruction, and I'm taking Calculus and this Physics class at the same time so I haven't learned any methods for this.

Ok, not unusual that you need calculus methods in physics before you get taught them in maths.
Do you know how to differentiate x^k with respect to x?

 

[IsoGD]

Ok, not unusual that you need calculus methods in physics before you get taught them in maths.
Do you know how to differentiate x^k with respect to x?

Well, after a bit of looking things up, I found out how to. I got the derivative (24 / (5*t^1/5)) - 2.75 after some trial and error. But after I found that and tried to solve it for where the slope equaled zero (like I read I was supposed to do?) I only found a time of t = .062 seconds, which is obviously wrong. What is it that I'm doing wrong? Am I not supposed to set the derivative equal to zero and solve for this problem and I'm actually supposed to something else?

 

[scottdave]

How did you get 0.062? Have you tried using logarithms to solve it rather than trial and error?

 

[IsoGD]

How did you get 0.062? Have you tried using logarithms to solve it rather than trial and error?

When I looked up how to do this, it said that I should set the derivative equal to zero, and solve for t. Every time I did that I got .062. I never saw anything on logarithms before.

 

[scottdave]

I see what you did, now. Did you take into account that t^1/5 is in the denominator? So logs are not the only way to solve it.

 

[IsoGD]

I see what you did, now. Did you take into account that t^1/5 is in the denominator? So logs are not the only way to solve it.

Yes, I did now. I took a fresh look at the equation again and tried to just re-solve from the beginning of the derivative being equal to zero, as it turns it out my issue was just some bad math at that step. I was able to get t = 16.2 seconds after I tried it again and it was correct on Webassign. And to double check, I plugged that back into the original equation and got m = 33.1 grams.

Now, I'm not sure how to do parts c and b for this question. I've tried plugging t = 2 into the original equation without the constant 22, and converting to kg/min but it's obviously not that easy, but that's all I could think to do with it.

 

[scottdave]

What you need to know about derivatives, since they are new to you: The derivative (with respect to time) is equal to the rate of change.
So if you have a function f(t) which is equal to grams, when you plug in seconds for time, then taking the derivative (which you have now done) will be an expression in grams/second. Plug in the specified times, then convert to the proper units (from grams/second to kg/minute)

 

[IsoGD]

What you need to know about derivatives, since they are new to you: The derivative (with respect to time) is equal to the rate of change.
So if you have a function f(t) which is equal to grams, when you plug in seconds for time, then taking the derivative (which you have now done) will be an expression in grams/second. Plug in the specified times, then convert to the proper units (from grams/second to kg/minute)

Wow, that explains a lot. Thank you! Thanks to that, I was able to get the two answers at .0857 kg/min and .0437 kg/min respectively for c and d, and they were correct on Webassign. I'm now finally done with this homework assignment because this was the last problem on it, so thanks for the help!

 

[scottdave]

Wow, that explains a lot. Thank you! Thanks to that, I was able to get the two answers at .0857 kg/min and .0437 kg/min respectively for c and d, and they were correct ...

Great! Do you understand how the derivative (rate) equaling zero is the place where a maximum (or minimum) will occur (for any general case)?

 

[IsoGD]

Great! Do you understand how the derivative (rate) equaling zero is the place where a maximum (or minimum) will occur (for any general case)?

Because the derivative (or rate) is essentially the slope of the equation, and when the slope (or rate) equals zero, the line is horizontal, meaning that it has hit a maximum or minimum, right?

 

[haruspex]

Because the derivative (or rate) is essentially the slope of the equation, and when the slope (or rate) equals zero, the line is horizontal, meaning that it has hit a maximum or minimum, right?

Yes.