Let's start with the original equation that Marta is trying to solve:
[tex]\[ S = 2 \pi h + 2 \pi r^2 \][/tex]
The goal is to solve for [tex]\( h \)[/tex].
First, we'll isolate the term that contains [tex]\( h \)[/tex]. To do that, we'll subtract [tex]\( 2 \pi r^2 \)[/tex] from both sides of the equation:
[tex]\[ S - 2 \pi r^2 = 2 \pi h \][/tex]
Next, we'll solve for [tex]\( h \)[/tex] by dividing both sides of the equation by [tex]\( 2 \pi \)[/tex]:
[tex]\[ h = \frac{S - 2 \pi r^2}{2 \pi} \][/tex]
Now, we can simplify the expression on the right-hand side:
[tex]\[ h = \frac{S}{2 \pi} - \frac{2 \pi r^2}{2 \pi} \][/tex]
By canceling out the [tex]\( 2 \pi \)[/tex] in the second term, we get:
[tex]\[ h = \frac{S}{2 \pi} - r^2 \][/tex]
Therefore, the correct result when solving the equation [tex]\( S = 2 \pi h + 2 \pi r^2 \)[/tex] for [tex]\( h \)[/tex] is:
[tex]\[ h = \frac{S}{2 \pi} - r^2 \][/tex]
Among the given choices, this matches with
[tex]\[ \frac{S}{2 \pi} - r^2 = h \][/tex]
None of the given options exactly match this result. However, if we rearrange none of the provided choices fit perfectly. This might indicate an issue with the provided options list.
Hence, none of the given choices for [tex]\( h \)[/tex] is correct based on the given criteria:
[tex]\[
\frac{S}{2\pi r} - r = h \quad(\text{Option 1})
\][/tex]
[tex]\[
\frac{S - r}{2\pi r} = h \quad(\text{Option 2})
\][/tex]
\]
\text{S} - \frac{r}{2 \pi} = h \quad(\text{Option 3})
\]
\]
\text{S} - \frac{2 \pi}{r} = h \quad(\text{Option 4})
\]
All of these forms are incorrect accordingly steps.
