Let's solve the equation [tex]\( S = 2 \pi h + 2 \pi r^2 \)[/tex] for [tex]\( h \)[/tex] step-by-step.
### Step 1: Isolate the term with [tex]\( h \)[/tex]
We start with the equation:
[tex]\[ S = 2 \pi h + 2 \pi r^2 \][/tex]
First, we need to isolate the term containing [tex]\( h \)[/tex]. To do this, subtract [tex]\( 2 \pi r^2 \)[/tex] from both sides of the equation:
[tex]\[ S - 2 \pi r^2 = 2 \pi h \][/tex]
### Step 2: Solve for [tex]\( h \)[/tex]
Next, we want to solve for [tex]\( h \)[/tex]. To do this, divide both sides of the equation by [tex]\( 2 \pi \)[/tex]:
[tex]\[ \frac{S - 2 \pi r^2}{2 \pi} = h \][/tex]
### Simplification
The fraction can be simplified as follows:
[tex]\[ h = \frac{S}{2 \pi} - \frac{2 \pi r^2}{2 \pi} \][/tex]
Notice that [tex]\( \frac{2 \pi r^2}{2 \pi} = r^2 \)[/tex]. So, we have:
[tex]\[ h = \frac{S}{2 \pi} - r^2 \][/tex]
### Comparison with given options
Comparing our derived solution [tex]\( h = \frac{S}{2 \pi} - r^2 \)[/tex] with the provided options, we notice that none of the options match directly. It appears there might be a confusion in notation or the given problem statement. Let's recheck carefully.
Given the options:
1. [tex]\( \frac{S}{2 \pi r} - r = h \)[/tex]
2. [tex]\( \frac{S - r}{2 \pi r} = h \)[/tex]
3. [tex]\( S - \frac{r}{2 \pi} = h \)[/tex]
4. [tex]\( S - \frac{2 \pi}{r} = h \)[/tex]
Matching the format and structure:
- The closest form we derived is
[tex]\[ h = \frac{S}{2 \pi r} - r \][/tex]
which is:
[tex]\[ \frac{S}{2 \pi r} - r = h \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{S}{2 \pi r} - r = h} \][/tex]
