Question

How do you solve for r in $ V = \pi {r^2}h $ ?

Answer 1

Hint: In order to solve the above formula of volume of cylinder for r first divide both sides by $ \pi h $ and consider the fact that the radius can never be a negative value to get your desired result.

Complete step-by-step answer:
We are given the volume of cylinder i.e. $ V = \pi {r^2}h $ where
r is the radius
h is the height of the cylinder
V denotes the Volume
So, to solve the formula of volume of cylinder $ V = \pi {r^2}h $ for r follow the steps below
 $ \Rightarrow V = \pi {r^2}h $
Divide both sides by $ \pi $
 $
   \Rightarrow \dfrac{V}{\pi } = \dfrac{{\pi {r^2}h}}{\pi } \\
   \Rightarrow \dfrac{V}{\pi } = \dfrac{{{\pi }{r^2}h}}{{{\pi }}} \\
   \Rightarrow \dfrac{V}{\pi } = {r^2}h \;
  $
Now dividing both by $ h $
 $ \Rightarrow \dfrac{V}{{\pi h}} = \dfrac{{{r^2}h}}{h} $
 $
   \Rightarrow \dfrac{V}{{\pi h}} = \dfrac{{{r^2}{h}}}{{{h}}} \\
   \Rightarrow \dfrac{V}{{\pi h}} = {r^2} \;
  $
Now taking $ \pm \sqrt {} $ on both the sides our formula now become
\[ \Rightarrow r = \pm \sqrt {\dfrac{V}{{\pi h}}} \]
Since, as we know that radius can never be a negative value
\[ \Rightarrow r = \sqrt {\dfrac{V}{{\pi h}}} \]
Therefore, the required solution of $ r $ is \[r = \sqrt {\dfrac{V}{{\pi h}}} \].
So, the correct answer is “\[r = \sqrt {\dfrac{V}{{\pi h}}} \]”.

Note: 1.The volume of a cylinder is the density of the cylinder which signifies the measure of material it can convey or how much measure of any material can be immersed in it. Cylinder's volume is given by the formula, πr2h, where r is the radius of the roundabout base and h is the stature of the cylinder. The material could be a fluid amount or any substance which can be filled in the cylinder consistently.
2.The measure of square units needed to cover the surface of the cylinder is the surface area of the cylinder. The formula for the surface area of the cylinder is equivalent to the complete surface area of the bases of the cylinder and the surface area of its sides.
 $ A = 2\pi {r^2} + 2\pi rh $