## Real World Applications – Algebra I

### Topic

How much better is using solar power for your home?

### Vocabulary

- Photovoltaic Effect: The phenomenon in which the incidence of light or other electromagnetic radiation upon the junction of two dissimilar materials, as a metal and a semiconductor the generation of an electromotive force (electricity!). In essence, it is the generation of electricity through the exposure of material to light. Photo = light, voltaic = electricity
- Photovoltaic Cell (Solar Cell): A thin semiconductor wafer specially treated to form an electric field, positive on one side and negative on the other. When light energy strikes the PV cell, electrons are knocked loose from the atoms in the semiconductor material. If electrical conductors are attached to the positive and negative sides, forming an electrical circuit, the electrons can be captured in the form of an electric current (electricity). This electricity can then be used to power a load, such as a light or a tool.
- Solar Energy: Solar energy is the solar radiation from the sun’s rays that reach the earth. This energy can be converted into other forms of energy, such as heat and electricity. Passive solar energy can be exploited by positioning a house to allow sunlight to enter through the windows to help heat a space. Active solar energy involves the conversion of sunlight to electrical energy, especially in solar (photovoltaic) cells. Solar Panel: A panel designed to absorb the sun’s ray as a source of energy for generating electricity or heat.
- Watt: A basic unit of power, equivalent to one joule per second, corresponding to the power in an electric circuit in which the potential difference is one volt and the current one

### Student Exploration

In this exploration, students will be able to investigate the energy output of solar panel systems through different representations of quadratic functions.

Hook: Watch this video:

\begin{align*}**\end{align*} It is advised that students do Activities 1-3 in the link provided below before doing this activity. This will give students a solid foundation of what solar energy is and how it is used.

The problem: A small photovoltaic solar electric panel system has been installed at a new school. The daily energy generated by the system can be modeled by the function \begin{align*}y = 64 + 95x - 7.5x^2\end{align*}, where \begin{align*}x\end{align*} is the month and \begin{align*}y\end{align*} is power in kilowatt-hours.

Questions:

- What observations can you make about the equation? What do the numbers mean? What do any of the coefficients mean? What do the variables represent? What’s the input? The output?
- Create a table to represent this function.
- What are realistic values for \begin{align*}x\end{align*}?

- Create a graph (either by hand or on a computer or graphing calculator) to represent this function.
- What observations can you make about this solar panel system?
- Identify the vertex of the graph. What does the vertex represent?
- Does the shape of the graph open UP or DOWN?
- What is the maximum monthly energy output?
- What month will have the maximum energy output?
- In which month will the system produce zero energy output?
- Does your answer to the previous question make sense in real life? Why or why not?

- From your observations and calculations above, create a more realistic graph to represent this function.
- Identify specific months on the appropriate axis.
- Why does this solar panel system represent a quadratic relationship?

### Extension Investigation

(This question has been directly transferred from the resource cited below.)

The energy output of a huge 2-megawatt photovoltaic solar farm in Arizona can be modeled with the quadratic function \begin{align*}f(x) = 71332 +63662 - 4868x^2\end{align*}. (Note: \begin{align*}1,000 \ watts = 1 \ kilowatt\end{align*} and \begin{align*}1,000 \ kilowatts = 1 \ megawatt\end{align*}.)

a. What does the 71332 in the function represent in your graph/table?

b. What does the 71332 represent in real life in this problem?

c. Find the vertex of the graph of the function.

d. What is the maximum energy output of the solar farm?

e. In what two months is the energy output at about 250 megawatts per month?

f. What is the least amount of energy the system will produce?

g. The electricity demand per month of a community of 250 average American homes is approximately 230 megawatts. Between what months of the year can this large solar farm meet this community’s electricity demand?