To solve for [tex]\( P \)[/tex] given the formula for the surface area [tex]\( S \)[/tex] of the enclosed space of a container in the shape of a rectangular pyramid, we start with the equation:
[tex]\[ S = lw + 0.5 Ph \][/tex]
We want to isolate [tex]\( P \)[/tex]. Let's rearrange the formula step-by-step:
1. Subtract [tex]\( lw \)[/tex] from both sides of the equation to isolate the term involving [tex]\( P \)[/tex]:
[tex]\[ S - lw = 0.5 Ph \][/tex]
2. Divide both sides of the equation by [tex]\( 0.5 h \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{S - lw}{0.5 h} \][/tex]
This simplification gives us the formula for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{S - lw}{0.5 h} \][/tex]
Now, we compare this derived formula with the given options:
1. [tex]\( P = S = hw - 0.5 h \)[/tex]
2. [tex]\( P = S + w + 0.5 h \)[/tex]
3. [tex]\( P = \frac{S - lw}{0.5 \text{\$}} \)[/tex]
4. [tex]\( P = \frac{S}{w + 0.5 h} \)[/tex]
The third option is:
[tex]\[ P = \frac{S - lw}{0.5 \text{\$}} \][/tex]
Here, the correct interpretation of the variable [tex]\( 0.5 \text{\$} \)[/tex] in context is equivalent to [tex]\( 0.5 h \)[/tex]. Thus, the correct choice is:
[tex]\[ P = \frac{S - lw}{0.5 h} \][/tex]
This corresponds to the third option.
Hence, the answer is option 3.
