We have a cylinder with a surface area of 1850 square meters and a radius of 9 meters. We want to estimate the volume of the cylinder to the nearest whole number. Let's begin by visualizing what this cylinder could look like.
To find the volume, we use the formula for the volume of a cylinder. This requires us to know the radius r and the height h.
V=π r^2 h
We know the radius of the cylinder, but not the height. Let's begin by finding a equation for the height using the surface area and the radius. Then we can use it to calculate the volume.
Finding an Equation for the Height
A cylinder is made up of three pieces — the top, the bottom, and the side. Both the top and the bottom are circles and the side is a rectangle. The rectangle's width is equal to the height of the cylinder and its length is equal to the circumference of the circles.
The total surface area SA of a cylinder is the combined area of the circles A_\text{C} and the rectangle A_\text{R}.
\begin{gathered}
SA=2A_\text{C}+A_\text{R}
\end{gathered}
We know how to calculate the area of a circle.
\begin{gathered}
A_\text{C}=\pi r^2
\end{gathered}
We also know how to calculate the area of a rectangle. In this case, we can rewrite the length to be the height of our cylinder and the width is the circumference C of our circles. Remember that the circumference C of a circle is found by multiplying 2π r.
\begin{aligned}
A_\text{R} & ={\color{#FF00FF}{\ell}} {\color{#A800DD}{w}} \\
& ={\color{#FF00FF}{h}}{\color{#A800DD}{C}}\\
& ={\color{#FF00FF}{h}}({\color{#A800DD}{2\pi r}})
\end{aligned}
Now, we can rewrite the equation for the surface area of our cylinder and isolate h.
