A large mixing tank currently contains  gallons of water, into which  pounds of sugar have been mixed. A tap will open, pouring  gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of  pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after  minutes.

To find the concentration of sugar in the tank after \( t \) minutes, we need to model how the amount of sugar and water changes with time. Let’s walk through this problem carefully.

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### Variables:
- Let \( V_0 \) = Initial gallons of water in the tank.
- Let \( S_0 \) = Initial pounds of sugar in the tank.
- \( r_V \) = Rate of water flowing into the tank in gallons per minute.
- \( r_S \) = Rate of sugar added to the tank in pounds per minute.
- \( t \) = Time (in minutes).
- \( V(t) \) = Volume of water after \( t \) minutes.
- \( S(t) \) = Amount of sugar after \( t \) minutes.

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### Step 1: Expressing water and sugar content after \( t \) minutes
- **Water**: Initially, there are \( V_0 \) gallons, and water is added at a rate of \( r_V \) gallons per minute. So after \( t \) minutes:
  \[
  V(t) = V_0 + r_V \cdot t
  \]

- **Sugar**: Initially, there are \( S_0 \) pounds of sugar, and additional sugar is poured at \( r_S \) pounds per minute. So after \( t \) minutes:
  \[
  S(t) = S_0 + r_S \cdot t
  \]

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### Step 2: Finding the concentration (pounds per gallon)
The concentration \( C(t) \) is the ratio of the total sugar to the total water at time \( t \):
\[
C(t) = \frac{S(t)}{V(t)}
\]

Using the expressions for \( S(t) \) and \( V(t) \):
\[
C(t) = \frac{S_0 + r_S \cdot t}{V_0 + r_V \cdot t}
\]

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### Step 3: Final Answer (Formula for Concentration)
After \( t \) minutes, the concentration of sugar in the tank (in pounds per gallon) is:
\[
C(t) = \frac{S_0 + r_S \cdot t}{V_0 + r_V \cdot t}
\]

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### Interpretation:
This formula tells us how the sugar concentration evolves over time. If both water and sugar are added at the same rate, the concentration might stabilize. Otherwise, it will change depending on the relative rates.

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Would you like to proceed with some specific numbers or further clarifications? 

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### Related Questions:
1. What happens to the concentration as \( t \to \infty \)?
2. How does the concentration behave if the sugar rate \( r_S \) is zero?
3. What if the water in the tank is drained simultaneously with the inflow?
4. How do we handle cases where the inflow rate \( r_V \) varies over time?
5. What if the initial amount of sugar \( S_0 \) is zero?

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**Tip:** Always check the units (pounds per gallon) carefully when working with rates and concentrations.

A large mixing tank currently contains gallons of water, into which pounds of sugar have been mixed. A tap will open, pouring gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after minutes.

Explore how to calculate the concentration of sugar in a mixing tank over time with given inflow rates of water and sugar. This step-by-step solution involves algebra, rates of change, and concentration ratios, suitable for high school students in Grades 10-12.

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