To solve for the cylinder's height [tex]\( h \)[/tex] in terms of the surface area [tex]\( A \)[/tex] and the radius [tex]\( r \)[/tex], we start with the given formula for the surface area of a cylinder:
[tex]\[ A = 2 \pi r (r + h) \][/tex]
Our goal is to rearrange this equation to isolate [tex]\( h \)[/tex].
1. Start with the equation:
[tex]\[ A = 2 \pi r (r + h) \][/tex]
2. Divide both sides of the equation by [tex]\( 2 \pi r \)[/tex] to separate the term involving [tex]\( h \)[/tex]:
[tex]\[ \frac{A}{2 \pi r} = r + h \][/tex]
3. Subtract [tex]\( r \)[/tex] from both sides to isolate [tex]\( h \)[/tex]:
[tex]\[ \frac{A}{2 \pi r} - r = h \][/tex]
So the formula for [tex]\( h \)[/tex] in terms of [tex]\( A \)[/tex] and [tex]\( r \)[/tex] is:
[tex]\[ h = \frac{A}{2 \pi r} - r \][/tex]
Now we compare this result to the given options:
- Option A: [tex]\( h = r + \frac{A}{2r} \)[/tex] — Incorrect
- Option B: [tex]\( h = \frac{A}{2 \pi} \)[/tex] — Incorrect
- Option C: [tex]\( h = \frac{1}{2er} - r^2 \)[/tex] — Incorrect
- Option D: [tex]\( h = \frac{A}{2 \pi r} - r \)[/tex] — Correct
Therefore, the correct formula is:
[tex]\[ \boxed{h = \frac{A}{2 \pi r} - r} \][/tex]
So answer D is correct.
