Sure, let's break this down step by step to determine the annual percentage increase given the enrollment rate of 42% per decade.
1. Understanding the Problem:
- The enrollment is increasing at a rate of 42% per decade.
- We need to find the equivalent annual percentage increase.
2. Equating the Decade Rate to an Annual Rate:
- A decade consists of 10 years.
- We need to find an annual percentage rate such that over 10 years, the overall increase is equivalent to 42%.
3. Using Compound Growth Formula:
- The cumulative growth over 10 years at an annual rate [tex]\( r_{annual} \)[/tex] can be represented by the compound growth formula:
[tex]\[
(1 + r_{annual})^{10} = 1 + 0.42
\][/tex]
- Here, [tex]\( r_{annual} \)[/tex] is the annual growth rate and 0.42 represents the growth per decade (42% as a decimal).
4. Solving for Annual Growth Rate:
- To find [tex]\( r_{annual} \)[/tex], we need to solve the equation:
[tex]\[
(1 + r_{annual})^{10} = 1.42
\][/tex]
- Taking the 10th root of both sides:
[tex]\[
1 + r_{annual} = \sqrt[10]{1.42}
\][/tex]
- Subtracting 1 from both sides:
[tex]\[
r_{annual} = \sqrt[10]{1.42} - 1
\][/tex]
5. Converting to Percentage:
- Since [tex]\( r_{annual} \)[/tex] is in decimal form, we multiply it by 100 to convert it to a percentage.
6. Intermediate Calculations:
- Evaluate [tex]\( \sqrt[10]{1.42} \)[/tex].
- Subtract 1 to find [tex]\( r_{annual} \)[/tex].
- Convert to percentage and round the result to one decimal place.
7. Final Answer:
- After performing the necessary calculations, the yearly percentage increase is found to be approximately:
[tex]\[
3.6\%
\][/tex]
Hence, the yearly percentage increase rate is 3.6%.
