Question

How do you solve for h in the equation S = $2\pi rh+2\pi {{r}^{2}}$ ?

Answer 1

Hint: In the above question, we have to solve S = $2\pi rh+2\pi {{r}^{2}}$ and find the value in terms of h. For solving this equation we have to isolate h and then find its value. The given equation is the formula of the surface area of a cylinder. So let’s see how we can solve this problem.

Step by step solution:
We have to solve S = $2\pi rh+2\pi {{r}^{2}}$ for h.
At first, we will subtract $2\pi {{r}^{2}}$ from both sides of the equation to isolate h
 $\Rightarrow S-2\pi {{r}^{2}}=2\pi rh+2\pi {{r}^{2}}-2\pi {{r}^{2}}$
 $\Rightarrow S-2\pi {{r}^{2}}=2\pi rh+0$
After we simplify, we get,
 $\Rightarrow S-2\pi {{r}^{2}}=2\pi rh$
 Now, we will divide $2\pi r$ from both sides of the equation
 $\Rightarrow \dfrac{S-2\pi {{r}^{2}}}{2\pi r}=\dfrac{2\pi rh}{2\pi r}$
After we simplify, we get,
 $\Rightarrow \dfrac{S-2\pi {{r}^{2}}}{2\pi r}=h$
 $\Rightarrow h=\dfrac{S-2\pi {{r}^{2}}}{2\pi r}$

Therefore, the value of h after solving S = $2\pi rh+2\pi {{r}^{2}}$ is $h=\dfrac{S-2\pi {{r}^{2}}}{2\pi r}$.

Additional Information:
The above equation in the question is the formula of the surface area of a cylinder where r is the radius of the cylinder, h is the height of the cylinder, and the value of $\pi $ = 3.142 or $\dfrac{22}{7}$. Moreover, you can see that from the very beginning of the answer we tried to separate h from all the variables and constants. So, at first, we subtracted the equation with $2\pi {{r}^{2}}$ and then we divided the equation with $2\pi r$.

Note:
There is an alternative way of solving this problem, let’s see that as well.
_{$\Rightarrow S=2\pi rh+2\pi {{r}^{2}}$}
Solving the above equation, we get,
 $\Rightarrow h=\dfrac{S}{2\pi r}-\dfrac{2\pi {{r}^{2}}}{2\pi r}$
After solving the above expression, we will get,
$\Rightarrow h=\dfrac{S}{2\pi r}-\dfrac{{{r}^{2}}}{r}$
After solving the above expression, we will get,
$\Rightarrow h=\dfrac{S}{2\pi r}-r$