Certainly! Let's determine the total surface area of the solar panels the agriculture club owns, including the new solar panels Jessica is installing and the one they already have.
Given are the surface areas of the new solar panels:
[tex]\[
\begin{align*}
A1 &= 28c^3 + 15c^2 - 31c + 11, \\
A2 &= 28c^3 + 31c^2 + c - 11, \\
A3 &= 28c^3 + 23c^2 - 15c, \\
A4 &= 8c^2 + 16c - 11.
\end{align*}
\][/tex]
We need to find the sum of these areas.
First, let's add up all the terms involving \( c^3 \):
[tex]\[
28c^3 + 28c^3 + 28c^3 = 84c^3.
\][/tex]
Next, let's add up all the terms involving \( c^2 \):
[tex]\[
15c^2 + 31c^2 + 23c^2 + 8c^2 = 77c^2.
\][/tex]
Now, let's add up all the terms involving \( c \):
[tex]\[
-31c + c - 15c + 16c = -29c.
\][/tex]
Finally, let's add up the constant terms:
[tex]\[
11 - 11 + 0 - 11 = -11.
\][/tex]
Putting all these together, the new total surface area is:
[tex]\[
84c^3 + 77c^2 - 29c - 11.
\][/tex]
This result accounts for the combined areas of all four solar panels. Let's include the one solar panel the agriculture club already has:
Given the surface area of the existing solar panel as \(8c^2 + 16c - 11,\) we add this to the total surface area calculated for the new panels to get the total surface area of all the solar panels the club owns.
Therefore, integrating the existing panel’s area does not alter this because it was originally considered within the calculation of \(A4\) already.
Hence, the total surface area of the solar panels is:
[tex]\[
84c^3 + 77c^2 - 29c - 11.
\][/tex]
