Solar Power The capacity of solar power installed in the...

Solar Power The capacity of solar power installed in the United States has grown since $2000 .$ The capacity of solar power installed in the United States can be modeled as $m(x)=15.45\left(1.46^{x}\right)$ megawatts between 2000 and 2008 , where $x$ is the number of years since 2000 . (Source: Based on data from USA Today, page $1 \mathrm{~B}$, $1 / 13 / 2009)$ a. How much solar power capacity was installed in 2000 , in $2004,$ in $2008 ?$ b. Use the information in part $a$ to make a table showing the year as a function of the capacity of solar power installed. c. Write a model giving the year as a function of the capacity of solar power installed.

Solar Power The capacity of solar power installed in the United States has grown since $2000 .$ The capacity of solar power installed in the United States can be modeled as
$m(x)=15.45\left(1.46^{x}\right)$ megawatts
between 2000 and 2008 , where $x$ is the number of years since 2000 . (Source: Based on data from USA Today, page $1 \mathrm{~B}$, $1 / 13 / 2009)$
a. How much solar power capacity was installed in 2000 , in $2004,$ in $2008 ?$
b. Use the information in part $a$ to make a table showing the year as a function of the capacity of solar power installed.
c. Write a model giving the year as a function of the capacity of solar power installed.

Key Concepts

Exponential Growth Model

An exponential growth model is used to represent phenomena that increase at a rate proportional to their current value. In these models, an initial value is multiplied by a constant growth factor raised to a power, which is often a function of time. This concept is crucial in understanding how small changes compound over time, leading to significant increases in value, as seen in various real-world applications such as population growth, investments, and energy capacity installations.

Independent and Dependent Variables

In the context of modeling real-world phenomena, independent and dependent variables play key roles. The independent variable is the one that is manipulated or considered as the input (in this case, the number of years since a base year), while the dependent variable is the output (such as the installed solar power capacity). Recognizing which variable influences the other helps in setting up the equation correctly and understanding how changes in one variable may affect the other.

Inverse Functions in Modeling

An inverse function reverses the roles of the input and output of a given function, making it possible to express one variable in terms of the other. In this scenario, inverse functions allow us to represent the year as a function of the capacity installed, rather than capacity as a function of the year. This is particularly useful when the goal is to determine the time at which a specific capacity was reached by solving the equation for the independent variable.

Data Representation Using Tables

Tables are often used in mathematical modeling to organize and display the relationship between variables in a clear and systematic way. Creating a table that shows the year as a function of capacity helps to verify the model, check for accuracy, and visualize the growth trend. By listing out corresponding pairs of years and capacities, one can better understand and communicate the progression of the modeled phenomenon over time.

Transcript

Please give Ace some feedback