Answer to Question #90748 in Geometry for Michael

Question

The area of a cylinder is 4m2 and it’s height is 0.9m determine the radius to 4

Answer 1

Surface area of a cylinder is the sum of areas of top "end", bottom "end" and the surface area of the side surface. Each "end" of a cylinder is made of a circle, so the surface area of each is

$$S_{end}=\pi r^2,$$

where $r$ is the radius of the "end". Area of "top" and "bottom" combined is

$$S_{ends}=2\pi r^2.$$

Side area of a cylinder can be found, knowing the cylinder without "ends" is "unrolled" as a rectangle. You can imagine this having a piece of paper and then rolling it to make a "pretend telescope".

Area of the side surface is the area of a rectangle with sides being a circumference and the height

$$S_{side}=2\pi r h,$$

where $r$ is the radius and $h$ is the height of the cylinder.

Then formula for the surface area of a cylinder becomes

$$S_{total}=2\pi r^2 + 2\pi r h.$$

So,

$$S_{total}=4m^2=2\pi(r^2+0.9r).$$

This is the quadratic equation relative to radius $r$ , so

$$r^2+0.9r-4/(2\pi)=0,$$

and radius is $r$ here.

$$r^2+0.9r−4/(2×3.141593)=0 \\ r^2+0.9r−0.63662=0$$

We then use quadratic formula with $a=1, b=0.9, c=-0.63662.$

$$r_{1,2}=−b\pm\frac{\sqrt{b^2−4ac}}{2a}$$

$$r_{1,2}=−0.9\pm\frac{\sqrt{0.9\times2−4\times 1\times(−0.63662)}}{2\times1}$$

$$r_{1,2}=-0.9\pm \sqrt{3.3564792}$$

$$r_{1,2}={0.46603480958290083;−1.366034809582901}$$

So, as radius is a nonnegative value, then radius is 0.4660m (up to 4 digits ).