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When we talk about a **constant percentage change**, we are referring to how much something grows or shrinks every year by a fixed percentage. In the context of models like those that describe the growth of solar power, this concept helps us understand how quickly things can change.
Take for example the formula given in the exercise: \[ M(x) = 15.45 \times 1.46^x \] This is an exponential growth model where "1.46" is a key number called the growth factor. To gauge how much percentage increase this signifies each year, use the formula: \[ \text{Constant Percentage Change} = (\text{Growth Factor} - 1) \times 100\% \] Substituting the growth factor from the model: \[ (1.46 - 1) \times 100\% = 46\% \] This means that there is a 46% increase in the amount of solar power capacity being installed annually.
This type of percentage change is extremely helpful in forecasting how fast things can escalate over time, especially in fields like renewable energy.

The **growth factor** is a crucial part of any exponential model. It tells us by what proportion the subject of the model increases each time period. In the solar power model we've been given: \[ M(x) = 15.45 \times 1.46^x \] The number "1.46" is the growth factor. It indicates that every year, the solar power capacity is multiplied by 1.46.
Breaking down its significance: - **Greater than 1**: Signals growth. Each year you're adding on more than was there before. For example, a growth factor of 1.10 means a 10% increase per year.
- **Equal to 1**: No growth occurs, the amount remains constant over time.
- **Less than 1**: Indicates decline, where each year less remains than the previous year.
In essence, the growth factor gives us a simplistic yet potent way to measure how fast things are either progressing or regressing over a period.

A **solar power model** is a mathematical representation that helps predict how the installation or capability of solar power changes over time. In our case, the model is:
\[ M(x) = 15.45 \times 1.46^x \] This specific model is crafted for estimating the installation growth of solar power from 2000 onward in the United States. Here's how it works:
- **Initial Value**: The number "15.45" represents the initial quantity or base amount of solar power installed at the start year (2000).
- **Exponential Growth**: The formula showcases how solar power capacity is not just rising, but rising exponentially, meaning growth happens at an increasingly rapid pace over time.
Understanding these models is crucial as they provide invaluable insights into future trends in solar energy. They help governments, companies, and consumers make informed decisions about adopting and investing in solar technology. Such models forecast when solar power might become the dominant energy source, given sustained growth patterns.