$A=2 \pi r^{2}+2 \pi r h$ $\newline$ The surface area, $A$ , of a cylinder of radius, $r$ , and height, $h$ , can be found with the given equation. Which of the following correctly shows the cylinder's height in terms of its radius and surface area? $\newline$ Choose $1$ answer: $\newline$ (A) $h=r-\frac{A}{2 \pi r}$ $\newline$ (B) $h=\frac{A}{2 \pi r}-r$ $\newline$ (C) $h=r-\frac{A}{r}$ $\newline$ (D) $h=\frac{A}{r}-r$

Full solution

A=2 \pi r^{2}+2 \pi r h

\newline

The surface area,

A

, of a cylinder of radius,

r

, and height,

h

, can be found with the given equation. Which of the following correctly shows the cylinder's height in terms of its radius and surface area?

\newline

Choose

1

answer:

\newline

(A)

h=r-\frac{A}{2 \pi r}

\newline

(B)

h=\frac{A}{2 \pi r}-r

\newline

(C)

h=r-\frac{A}{r}

\newline

(D)

h=\frac{A}{r}-r

Given Surface Area Formula: We are given the surface area formula for a cylinder: $A = 2\pi r^2 + 2\pi rh$ . We need to solve for $h$ in terms of $r$ and $A$ .
Isolate Term with h: First, subtract the term $2\pi r^2$ from both sides of the equation to isolate the term with $h$ on one side. This gives us $A - 2\pi r^2 = 2\pi rh$ .
Solve for h: Next, divide both sides of the equation by $2\pi r$ to solve for $h$ . This gives us $\frac{A - 2\pi r^2}{2\pi r} = h$ .
Simplify Right Side: Simplify the right side of the equation by splitting the fraction into two parts: $\frac{A}{2\pi r} - \frac{2\pi r^2}{2\pi r} = h$ .
Final Expression for Height: Simplify the second term of the right side of the equation: $(2\pi r^2) / (2\pi r)$ simplifies to $r$ , because the $2\pi r$ in the numerator and denominator cancel out, and $r^2 / r$ simplifies to $r$ . This gives us $A / (2\pi r) - r = h$ .
Final Expression for Height: Simplify the second term of the right side of the equation: $(2\pi r^2) / (2\pi r)$ simplifies to $r$ , because the $2\pi r$ in the numerator and denominator cancel out, and $r^2 / r$ simplifies to $r$ . This gives us $A / (2\pi r) - r = h$ .The final expression for the height ( $h$ ) of the cylinder in terms of its radius ( $r$ ) and surface area ( $A$ ) is $h = (A / (2\pi r)) - r$ , which corresponds to option (B).