In​ 2007, a country had a population of 82 comma 700 thousand. At that​ time, the​ country's population was declining at a rate of 0.032​% per year. If this​ continues, how long will it take the​ country's population to reach​ 82,000 thousand? Use y equals y 0 e Superscript negative 0.00032 t.
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Part 1
It will take approximately 
  
enter your response here years.

To solve this, let's use the exponential decay formula:

\[
y = y_0 e^{-kt}
\]

where:
- \( y_0 = 82,700 \) thousand (the initial population),
- \( y = 82,000 \) thousand (the target population),
- \( k = 0.00032 \) (the decay rate),
- \( t \) is the time in years we are trying to find.

Let's start by rearranging the formula to solve for \( t \):

1. Substitute the values into the formula:

   \[
   82,000 = 82,700 e^{-0.00032t}
   \]

2. Divide both sides by 82,700 to isolate the exponential term:

   \[
   \frac{82,000}{82,700} = e^{-0.00032t}
   \]

3. Calculate the left side:

   \[
   \frac{82,000}{82,700} \approx 0.9915
   \]

4. Now, take the natural logarithm of both sides to solve for \( t \):

   \[
   \ln(0.9915) = -0.00032t
   \]

5. Calculate \( \ln(0.9915) \):

   \[
   \ln(0.9915) \approx -0.0085
   \]

6. Now, divide by \(-0.00032\) to find \( t \):

   \[
   t = \frac{-0.0085}{-0.00032} \approx 26.56
   \]

So, it will take approximately **26.56 years** for the population to reach 82,000 thousand.

Would you like a more detailed explanation, or do you have any questions?

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### Related Questions
1. How would the time change if the decay rate was 0.05% instead of 0.032%?
2. What is the population after 10 years with the same decay rate?
3. If the population started at 100,000 thousand, how long would it take to reach 82,000 thousand?
4. How would the equation change if the population was increasing instead of decreasing?
5. What is the half-life of this population decline at a rate of 0.032% per year?

### Tip
Remember, when solving for \( t \) in exponential decay, taking the natural logarithm is essential to isolate the variable in the exponent.

In 2007, a country had a population of 82,700 thousand. At that time, the country's population was declining at a rate of 0.032% per year. If this continues, how long will it take the country's population to reach 82,000 thousand? Use y = y₀ e^(-0.00032t).

Calculate the time required for a population to decrease from 82,700 thousand to 82,000 thousand using exponential decay at a 0.032% annual rate. Learn the steps to apply logarithmic properties and exponential decay functions. Ideal for students in Grades 10-12.

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