Question

The number of students enrolled at a college is 16 comma 000 and grows 4​% each year. Complete parts ​(a) and ​(b).
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Part 1
​a) Complete the equation yequalsnothing times nothing Superscript nothing
  
  
  
to find the number of students enrolled after x years.
​b) Use your equation to predict the number of students enrolled after 24 years.
After 24 ​years, there will be
  
enter your response here students enrolled.
​(Round to the nearest whole number as​ needed.)

Answer 1

To find the number of students enrolled at the college after \( x \) years when it grows at a rate of 4% annually, we can use the formula for exponential growth:

\[
y = P(1 + r)^x
\]

Where:
- \( y \) is the future value (the number of students after \( x \) years),
- \( P \) is the initial amount (the number of students currently enrolled),
- \( r \) is the growth rate (expressed as a decimal), and
- \( x \) is the number of years.

### Part 1
**a)** Here, we identify:
- \( P = 16000 \) (the initial enrollment)
- \( r = 0.04 \) (4% growth rate)

So the equation becomes:

\[
y = 16000(1 + 0.04)^x
\]

This simplifies to:

\[
y = 16000(1.04)^x
\]

### Part 2
**b)** Now we will predict the number of students enrolled after 24 years.

Substituting \( x = 24 \) into the equation:

\[
y = 16000(1.04)^{24}
\]

Calculating \( (1.04)^{24} \):

\[
(1.04)^{24} \approx 2.566
\]

So now:

\[
y = 16000 \times 2.566 \approx 41056
\]

Thus, rounding to the nearest whole number, the prediction for the number of students enrolled after 24 years is:

\[
\text{After 24 years, there will be approximately } 41056 \text{ students enrolled.}
\]

If you need any further calculations or clarifications, feel free to ask!