Question

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

Answer 1

Hint: We need to isolate $h$ from the equation and bring it to the left side and keep all the other terms on the right side. We first subtract $2\pi {r^2}$ from both sides. And then divide both sides with $2\pi r$ to isolate $h$ and get our required answer.

Complete step-by-step solution:
Given equation is: $S = 2\pi rh + 2\pi {r^2}$, from here we need to solve for $h$. Thus we need to isolate the same to one side of the equation and have all other variables to the other side of the equation.
First we subtract $2\pi {r^2}$ from both the sides of the equation to remove it from the right hand side:
$ \Rightarrow S - 2\pi {r^2} = 2\pi rh + 2\pi {r^2} - 2\pi {r^2}$
On solving the right hand side of the equation, we get:
$ \Rightarrow S - 2\pi {r^2} = 2\pi rh$
Now we divide both sides of the equation given above with $2\pi r$ to isolate our required variable $h$ :
\[ \Rightarrow \dfrac{{S - 2\pi {r^2}}}{{2\pi r}} = \dfrac{{2\pi rh}}{{2\pi r}}\]
On solving the right hand side of the equation, we get:
$ \Rightarrow \dfrac{{S - 2\pi {r^2}}}{{2\pi r}} = h$
We have successfully isolated $h$ on the right hand side of our equation. We need to solve the left hand side a bit further. We separate the numerator into different terms, keeping the denominator common:
$ \Rightarrow \dfrac{S}{{2\pi r}} - \dfrac{{2\pi {r^2}}}{{2\pi r}} = h$
We solve the left hand side further, thus we get:
$ \Rightarrow \dfrac{S}{{2\pi r}} - r = h$

Thus, we have successfully solved for $h$:
$ h = \dfrac{S}{{2\pi r}} - r$

Note: Alternate method of solving:
The equation is given as: $S = 2\pi rh + 2\pi {r^2}$
On the right hand side, we take out the common factor from all the variables so that we can isolate out $h$.
In the given equation, $2\pi r$ is common to both the terms on the right hand side. The equation now becomes:
$ \Rightarrow S = 2\pi r\left( {h + r} \right)$
Now let’s divide both left hand side and right hand side with $2\pi r$, thus we get:
\[ \Rightarrow \dfrac{S}{{2\pi r}} = \dfrac{{2\pi r\left( {h + r} \right)}}{{2\pi r}}\]
On solving the right hand side further, we get:
\[ \Rightarrow \dfrac{S}{{2\pi r}} = h + r\]
Now let’s subtract r from both the sides, we get:
$ \Rightarrow \dfrac{S}{{2\pi r}} - r = h$