Final answer:
Exponential models are used to describe growth or decay, where factors such as the base and exponent represent the rate and duration of change. When assessing growth over time, logarithms can be used to simplify calculations for small growth rates. In linear equations involving tax rates, algebra is used to solve for a single variable.
Step-by-step explanation:
The exponential models in terms of t provided by the student describe exponential growth or decay based on the base chosen. Specifically:
- i. 4000 * (1 + 0.07)t represents exponential growth at a rate of 7% annually.
- ii. 4000 * (1 - 0.07)t represents exponential decay at a rate of 7% annually.
- iii. 4000 * 1.07t is another way of writing exponential growth at a 7% rate, with the growth factor directly indicated as 1.07t.
- iv. 4000 * 0.93t mirrors the decay model with a base value that directly indicates a 7% decrease per period.
When assessing growth over time, equations similar to these are applied. For example, to evaluate growth in 10 years at a continuous growth rate (p), you can use t2 = ln 2/ln(1 + p). If p is small, such as 2%, the rate r in the equation r = ln(1 + p) can be approximately equal to p for simplicity.
To solve for a variable like Y given a linear equation that includes a tax rate T, you would reform the equation to have a single variable and solve algebraically, as exemplified by the provided steps.