Problem Overview

The question requires calculating the average annual growth rate of bottle production over a 2-year period, given the initial and final production numbers.

• Initial Production ($P_0$): $16000$
• Final Production ($P_n$): $26000$
• Number of Years ($n$): $2$
• Objective: Find the annual growth rate ($r$).

Calculating Annual Growth Rate

The Compound Annual Growth Rate (CAGR) formula is used to determine the average yearly growth:

$ P_n = P_0 \times (1 + r)^n $

To find the rate $r$, we rearrange the formula:

$ r = \left(\frac{P_n}{P_0}\right)^{\frac{1}{n}} - 1 $

Step-by-Step Calculation

1. Calculate the production ratio: Divide the final production by the initial production.

   $ \text{Ratio} = \frac{26000}{16000} = \frac{13}{8} = 1.625 $

2. Calculate the annual growth factor: Raise the ratio to the power of $1/n$.

   $ \text{Growth Factor} = (1.625)^{\frac{1}{2}} $

   $ \text{Growth Factor} \approx 1.274754 $

3. Determine the annual growth rate ($r$): Subtract 1 from the growth factor and express it as a percentage.

   $ r = \text{Growth Factor} - 1 $

   $ r \approx 1.274754 - 1 = 0.274754 $

   $ r \approx 0.274754 \times 100\% \approx 27.47\% $

The rate of growth per annum is approximately $27.47\%$.