The Oakdale Opera House sold 4,307 tickets this year, but it has been experiencing a 9% drop in sales from one year to the next. If this trend continues, then how many tickets will... The Oakdale Opera House sold 4,307 tickets this year, but it has been experiencing a 9% drop in sales from one year to the next. If this trend continues, then how many tickets will the opera house sell 9 years from now? If necessary, round your answer to the nearest whole number. Read more

Steps to Solve

1. Identify initial values and the decay rate

The initial number of tickets sold is 4,307. The annual decay rate is 9%, which we express as a decimal: $$ r = 0.09 $$

2. Calculate the decay factor

To find the decay factor, subtract the decay rate from 1: $$ \text{decay factor} = 1 - r = 1 - 0.09 = 0.91 $$

3. Apply the exponential decay formula

The formula for exponential decay is given by: $$ N = N_0 \times (decay , factor)^t $$ Where:

• $N_0$ is the initial value (number of tickets sold now),
• $t$ is the number of years (9 years in this case).

Substituting our values: $$ N = 4307 \times (0.91)^9 $$

4. Calculate the number of tickets sold after 9 years

Now calculate ( (0.91)^9 ): $$ (0.91)^9 \approx 0.42241 $$

Then calculate: $$ N \approx 4307 \times 0.42241 \approx 1827.04467 $$

5. Round to the nearest whole number

Rounding 1827.04467 gives us 1827.