A 300-liter tank initially full of water develops a leak at the bottom. Given that 20% of the water leaks out in the first 17 minutes, give a formula for the amount of water left in the tank t
 minutes after the leak develops if the water drains off at a rate that is proportional to the amount of water present.

To solve this problem, we will model the situation using a differential equation since the rate of water draining is proportional to the amount of water left in the tank at any given time.

### Step-by-Step Solution:

Let:
- \( W(t) \) be the amount of water in the tank at time \( t \) (in minutes),
- \( W(0) = 300 \) liters, the initial amount of water,
- The rate of water draining is proportional to the amount of water present, i.e., \( \frac{dW}{dt} = -kW(t) \), where \( k \) is a constant of proportionality.

#### 1. Formulate the differential equation
The rate of water draining is proportional to the amount of water, so the equation becomes:

\[
\frac{dW}{dt} = -kW(t)
\]

#### 2. Solve the differential equation
This is a separable differential equation. Rearranging terms and integrating:

\[
\frac{1}{W(t)} dW = -k dt
\]

Integrating both sides:

\[
\ln |W(t)| = -kt + C
\]

Exponentiating both sides:

\[
W(t) = e^{-kt + C} = e^C e^{-kt}
\]

Let \( e^C = W(0) = 300 \), the initial condition, so the solution becomes:

\[
W(t) = 300 e^{-kt}
\]

#### 3. Use the given information to find \( k \)
We are told that 20% of the water leaks out in the first 17 minutes. This means that after 17 minutes, 80% of the water remains, i.e., \( W(17) = 0.8 \times 300 = 240 \) liters.

Substituting into the equation:

\[
240 = 300 e^{-17k}
\]

Solving for \( k \):

\[
\frac{240}{300} = e^{-17k}
\]

\[
0.8 = e^{-17k}
\]

Taking the natural logarithm of both sides:

\[
\ln(0.8) = -17k
\]

\[
k = -\frac{\ln(0.8)}{17}
\]

Using the value of \( \ln(0.8) \approx -0.2231 \), we find:

\[
k \approx \frac{0.2231}{17} \approx 0.01312
\]

#### 4. Final formula for the amount of water
Now that we know \( k \), we can substitute it back into the equation for \( W(t) \):

\[
W(t) = 300 e^{-0.01312t}
\]

This is the formula for the amount of water left in the tank at time \( t \) minutes after the leak develops.

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Would you like to see any details of this process, or do you have further questions?

Here are five related questions you might find interesting:
1. How would the equation change if 30% of the water leaked out in the first 17 minutes instead of 20%?
2. What happens to the amount of water after a long period of time (i.e., as \( t \to \infty \))?
3. How would the solution change if the rate of leakage were not proportional to the water present but to the square of the water present?
4. How long would it take for 50% of the water to leak out?
5. How can you verify if the draining follows an exponential model in real-life situations?

**Tip:** When solving exponential decay problems, always check for key time points (e.g., when the volume decreases by half or another known percentage) to estimate constants accurately.

A 300-liter tank initially full of water develops a leak at the bottom. Given that 20% of the water leaks out in the first 17 minutes, give a formula for the amount of water left in the tank t minutes after the leak develops if the water drains off at a rate that is proportional to the amount of water present.

Solve a differential equation for water leakage in a 300-liter tank, where 20% of the water leaks out in 17 minutes. Learn how to model the situation with exponential decay and find a formula for the water remaining after t minutes. Suitable for Grades 11-12.

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