To solve this problem, let's break it down step-by-step:
1. Understand the initial conditions and growth rate:
- The initial number of toys produced each year is [tex]\(1,250,000\)[/tex].
- This quantity increases by 150% each year.
2. Convert the initial quantity into millions:
- Since [tex]\(1,250,000\)[/tex] toys are produced, in terms of millions, this is [tex]\( \frac{1,250,000}{1,000,000} = 1.25 \)[/tex] million toys.
3. Determine the growth factor:
- The increase of 150% per year means the number of toys grows by 1.5 times the initial quantity each year, in addition to the initial quantity itself.
- Thus, adding the initial quantity, the growth factor is [tex]\( 1 + 1.5 = 2.5 \)[/tex].
4. Formulate the exponential growth model:
- In exponential growth, the quantity increases by a constant factor each year.
- The general form of an exponential growth equation is [tex]\( n = n_{\text{initial}} \times (\text{growth factor})^t \)[/tex], where [tex]\( t \)[/tex] is the number of years.
- Here, [tex]\( n_{\text{initial}} \)[/tex] is [tex]\(1.25\)[/tex] million and the growth factor is [tex]\(2.5\)[/tex].
Thus, substituting the values:
[tex]\[ n = 1.25 \cdot 2.5^t \][/tex]
5. Evaluate the given options:
- [tex]\( n=\frac{2.5(1.5)}{t}, t \neq 0 \)[/tex] is not a correct format and doesn’t properly represent exponential growth.
- [tex]\( n=1.5 t^2+1.25 \)[/tex] represents quadratic growth, not suitable for our scenario.
- [tex]\( n=1.5 t+1.25 \)[/tex] represents linear growth, not suitable for exponential increase.
- [tex]\( n=1.25 \cdot 2.5^t \)[/tex] represents exponential growth and matches our derived formula. This is the correct model.
Therefore, the correct model to find the number of toys, [tex]\( n \)[/tex] (in millions), being produced in [tex]\( t \)[/tex] years is:
[tex]\[ n = 1.25 \cdot 2.5^t \][/tex]
