21.Application: A solar panel is used to power an overhead sign on a highway. Ench square metre of solar panel receives about 200 W of solar power. This solar panel converts about 15% of the solar energy to electrical power. (a) Write a function for the solar power, S(A) measured in watts (W) for any given area, A in square metres. (b) Write a function for the total electrical power. P(S) in watts (W) this solar panel can generate. (c) How large must the solar panel be if the sign uses 6 W of electrical energy?

## Step 1: Define the solar power function<br />### The solar power, $S(A)$, is 200 W per square meter. Therefore, for an area $A$ in square meters, the total solar power received is $S(A) = 200A$ watts.<br />## Step 2: Define the electrical power function<br />### The solar panel converts 15% of the solar energy to electrical power. So, if $S$ is the solar power in watts, the electrical power $P(S)$ is given by $P(S) = 0.15S$.<br />## Step 3: Combine the functions and solve for the required area<br />### We can express the electrical power as a function of the area by substituting $S(A)$ into $P(S)$: $P(A) = P(S(A)) = 0.15 * 200A = 30A$. The sign requires 6W of electrical power, so we set $P(A) = 6$ and solve for $A$: $6 = 30A$, which gives $A = \frac{6}{30} = \frac{1}{5} = 0.2$ square meters.