Question

The number of psychiatry papers published each year decreases by 7%. if 26,000 papers were published in the current year, how many papers would be expected to be published in 5 years?.(round to nearest whole number)
find a in this equation using y=a(b)^x
find base
find exponential equation
find how many papers are published after 5 years

Answer 1

To solve the problem, we'll use the formula for exponential decay, which is given by the equation:

\[ y = a(b)^x \]

where:
- \( y \) is the amount after \( x \) years,
- \( a \) is the initial amount,
- \( b \) is the base,
- \( x \) is the number of years.

### Step 1: Identify \( a \) and \( b \)

1. **Initial amount \( a \)**: This is the number of papers published in the current year, which is given as 26,000. Therefore, \( a = 26,000 \).

2. **Base \( b \)**: Since the number of papers published decreases by 7% each year, we can express that as:

\[ b = 1 - 0.07 = 0.93 \]

### Step 2: Write the exponential equation

We can now construct the exponential equation:

\[ y = 26000(0.93)^x \]

### Step 3: Calculate the number of papers published after 5 years

Now, we need to find out how many papers will be published after \( x = 5 \) years. We substitute \( x = 5 \) into the equation:

\[
y = 26000(0.93)^5
\]

Calculating \( (0.93)^5 \):

\[
(0.93)^5 \approx 0.6593
\]

Now substituting this back into the equation:

\[
y \approx 26000 \times 0.6593 \approx 17140.78
\]

### Step 4: Round to the nearest whole number

Finally, rounding to the nearest whole number:

\[
y \approx 17141
\]

### Summary

- **Initial amount (\( a \))**: 26,000
- **Base (\( b \))**: 0.93
- **Exponential equation**: \( y = 26000(0.93)^x \)
- **Number of papers published after 5 years**: **17,141**.