Exponential growth: ""Grain is pouring from a hopper on to a barn floor where it forms a conical pile..."

A

Aminta_1900

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I'm having a little difficulty with this question. The many examples I've seen before this are mostly in the form y=y[o]e^kt, and naturally involve solving for k or t with the variables given, or vice versa. This below is the first question and doesn't involve e or give numerical values for any of the variables, so I'm a bit stumped in how to use C and the initial value.

"Grain is pouring from a hopper on to a barn floor where it forms a conical pile whose height h is increasing at a rate that is inversely proportional to h^3. The initial height of the pile is h[0] and the height doubles after time T. Find, in terms of T, the time after which the height has grown to 3h[0]."

My workings...

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Thanks.
 

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In line 2, you're assuming C = 0; why?

In line 4, you're assuming C = h_0; why?

Why not first solve for C from line 1, using the fact that h = h_0 at t = 0?
 
Dr.Peterson said:
In line 2, you're assuming C = 0; why?

In line 4, you're assuming C = h_0; why?

Why not first solve for C from line 1, using the fact that h = h_0 at t = 0?
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I first tried without the C because one of the examples in my textbook also ignored C.

Using h_0 for C was just a hunch, since in y = Ce^kt, C would be y_0, but I wasn't really hopeful of this!
 
OK, I'm sure I have now solved for C, but I'm still getting a wrong answer, so somewhere I am making a mistake. If anything could point this out for me....

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The answer should be (16/3)T.
 
Just for completeness, y=y[o]e^kt and y=y[o]e^(kt) are not the same!

\(\displaystyle y_0e^{k}t\neq y_0e^{kt}\) You really need to put those parenthesis around the entire power!
 
Aminta_1900 said:
I'm having a little difficulty with this question. The many examples I've seen before this are mostly in the form y=y[o]e^kt, and naturally involve solving for k or t with the variables given, or vice versa. This below is the first question and doesn't involve e or give numerical values for any of the variables, so I'm a bit stumped in how to use C and the initial value.
Click to expand...
It may also be helpful to comment on your misunderstanding of what this is.

It is not exponential growth; that can always be written as [imath]y=y_0e^{kt}[/imath]. This is a different differential equation, in which the rate, rather than increasing, actually decreases. So comparing the problem to exponential growth may have led you astray.

On the other hand, the work here is very similar to that involved in exponential growth; if you have seen problems where you were given half-life or doubling time, you were not given specific values for the variables, but will have taken the initial value as an unknown parameter, as you did here. And, as here, you found C by using the given values, not by guessing. So there is a lot of similarity.
 
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