I would greatly appreciate your help with a math equation needed for a science experiment I'm researching. The case is the following: there is a pit full of juice and a stream of water is flowing through the pit (so that the amount of water which enters also exits). The goal is for there to be only 3 units of juice in the pit (regardless of the amount of water). As the stream flows through we assume that the stream water mixes evenly with the juice in the pit so that what exits is a proportion of stream water and juice. The question I'm stuck on is, how much stream water needs to pass through the pit in order for there to remain only 3 units of juice water. Since the proportions constantly change, it seems like a calc question to me, although I may be wrong. Could you do me a favor and email me the equation that I could use for any given amount of juice?

Question

I would greatly appreciate your help with a math equation needed for a science experiment I'm researching. The case is the following: there is a pit full of juice and a stream of water is flowing through the pit (so that the amount of water which enters also exits). The goal is for there to be only 3 units of juice in the pit (regardless of the amount of water). As the stream flows through we assume that the stream water mixes evenly with the juice in the pit so that what exits is a proportion of stream water and juice. The question I'm stuck on is, how much stream water needs to pass through the pit in order for there to remain only 3 units of juice water. Since the proportions constantly change, it seems like a calc question to me, although I may be wrong. Could you do me a favor and email me the equation that I could use for any given amount of juice?

Thanks,
Joseph
jadinoff @ gmail . com

Answer 1

You can set up a first order differential equation to determine the time at which the required concentration remains.

The volume, V of the pit should be known.
The initial amount of juice, j, should be known.

The input concentration c1 of juice is zero.
The input rate r1=r=Q/t (in m&sup3/s) should be known.

Assuming perfect agitation, the output concentration is c2=j/V.
The output rate, r2 equals input rate r1=r.

The equation would be

dj/dt = r1*c1-r2*c2

substituting values,
dj/dt = r*0 - r*(j/V)
dj/dt = -(Q/(Vt))j

Separate variables, to get
dj/j = -(Q/V)*dt/t
Integrate (do not forget the integration constant) and solve for j in terms of t.

Answer 2

Certainly, Joseph! I'd be happy to help you with the math equation for your science experiment. This problem can be solved using a simple proportion. Let's assume that the amount of stream water passing through the pit is represented by "x" (in some units).

To determine the proportion of stream water and juice that exits the pit, we need to consider the overall amount of liquid in the pit. Since the goal is to have only 3 units of juice remaining, the total amount of liquid in the pit should be x + 3 (units).

Now, if we assume that the stream water mixes evenly with the juice, the proportion of stream water in the total liquid will be x / (x + 3) and the proportion of juice will be 3 / (x + 3).

Since the problem states that the amount of water entering equals the amount of water exiting, we can set up the following equation:

x = (x / (x + 3)) * (x + 3)

To solve this equation, we can cross-multiply:

x = x

Therefore, this equation is true for any value of x.

This means that regardless of the amount of stream water passing through the pit, as long as the goal is to have 3 units of juice remaining, the equation x = x will always be true.

In other words, there is no specific equation to calculate the amount of stream water needed for any given amount of juice. The amount of stream water passing through the pit can be any value, as long as the initial amount of juice plus the amount of stream water entering is equal to the desired amount of 3 units of juice.