Surface Area — Question 10

Question 10

Given: A cylinder has radius $r$, height $h$, and fixed lateral area $A$.
Tasks: Solve for $h$, find the total area including both circular ends, and then analyze the limit as $r\to0^+$.

Question 10 – Solution

Step 1: Solve the lateral-area equation for $h$. $$\begin{align*} 2\pi rh&=A,\\ h&=\frac{A}{2\pi r}. \end{align*}$$ Step 2: Add the two circular ends. Each end has area $\pi r^2$, so $$S_{\mathrm{total}}=A+2\pi r^2.$$ Step 3: Take the limit as $r\to0^+$. $$\begin{align*} \lim_{r\to0^+}S_{\mathrm{total}} &=\lim_{r\to0^+}(A+2\pi r^2)=A,\\ \lim_{r\to0^+}h &=\lim_{r\to0^+}\frac{A}{2\pi r}=\infty. \end{align*}$$ $$\boxed{S_{\mathrm{total}}\to A,\qquad h\to\infty}$$