Question

Suppose you decide to cut the wire into two pieces (not necessarily the same length) to shape into two circles. Write a function A ( x ) which models the total area of the two circles in terms of x , the length of one of the pieces of wire. (It may be helpful to use for a circle A = πr 2 and C = 2 πr ). Then find the length x that will minimize the total area.

Answer 1

`2\pi r(r+rx)`

Step-by-step explanation:

Let the area of one circular side be given by the formula :
`A_(1) = \pi r^(2)`

However, the wire is a solid cylinder, then it means that the total area is 2 ×
`\pi r^(2)` =
`2\pi r^(2)`

However, there is the surface area to consider. This is the curved area of the wire. This is given as:

`A_(2) = lb`

The length is x.

The breadth is calculated as follows - the length of the circle =
`\pi D = 2\pi r`

Then the area = lb

=
`2\pi rx`

Therefore, the total area is given as
`A_(1) + A_(2)`

=
`2\pi r^(2) + 2\pi rx\\ 2\pi r(r+rx)`