Exponential model for Quickwater Public Library movie rentals after Bernard was hired — y = 207(0.9)^x (months until rentals < 100: 7 months)

Exponential equation:

y = 207(0.9)^x

Explanation: At x = 0 (Bernard's first month) there were 207 rentals, so a = 207. A 10% monthly decrease means each month we multiply by 1 - 0.10 = 0.90, so b = 0.9.

When will rentals be less than 100?

1. Set up the inequality: 207(0.9)^x < 100.
2. Divide both sides by 207: (0.9)^x < 100/207 ≈ 0.48309.
3. Take natural logs: x ln(0.9) < ln(100/207). Since ln(0.9) < 0, dividing by ln(0.9) reverses the inequality:
4. x > ln(100/207) / ln(0.9) ≈ 6.91.
5. So rentals drop below 100 after more than 6.91 months. Rounding up to whole months, that occurs after 7 months.

Quick check: y(6) = 207(0.9)^6 ≈ 110 > 100, and y(7) = 207(0.9)^7 ≈ 99 < 100, so 7 months is correct.