Question

1. enrollment in 2000, 2200 students attended polaris high school. the enrollment has been declining 2% annually. a. write an equation for the enrollment of polaris high school t years after 2000. b. if this trend continues, how many students will be enrolled in 2015?

Explanation:

Part (a)

Step 1: Identify the type of growth/decay

This is an exponential decay problem since the enrollment is declining. The general formula for exponential decay is \( y = a(1 - r)^t \), where \( a \) is the initial amount, \( r \) is the rate of decay (as a decimal), and \( t \) is time.

Step 2: Determine the values of \( a \) and \( r \)

• The initial enrollment (\( a \)) in 2000 is 2200 students.
• The annual decline rate is 2%, so \( r = 0.02 \) (since 2% = \( \frac{2}{100} = 0.02 \)).

Step 3: Write the equation

Substitute \( a = 2200 \) and \( r = 0.02 \) into the exponential decay formula:
\( y = 2200(1 - 0.02)^t \)
Simplify \( 1 - 0.02 = 0.98 \), so the equation is \( y = 2200(0.98)^t \), where \( y \) is the enrollment \( t \) years after 2000.

Step 1: Determine the value of \( t \)

We need to find the enrollment in 2015. Since 2015 is \( 2015 - 2000 = 15 \) years after 2000, \( t = 15 \).

Step 2: Substitute \( t = 15 \) into the equation from part (a)

The equation is \( y = 2200(0.98)^t \). Substitute \( t = 15 \):
\( y = 2200(0.98)^{15} \)

Step 3: Calculate \( (0.98)^{15} \)

Using a calculator, \( (0.98)^{15} \approx 0.7397 \) (you can calculate this by repeatedly multiplying 0.98 fifteen times or using the exponent function on a calculator).

Step 4: Calculate \( y \)

Multiply 2200 by 0.7397:
\( y \approx 2200 \times 0.7397 \approx 1627.34 \)
Since we can't have a fraction of a student, we round to the nearest whole number, so \( y \approx 1627 \).

Answer:

(a):
\( y = 2200(0.98)^t \)

Part (b)