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

Answer:

$2\pi r(r+rx)$

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 $A_{1} + A_{2}$

                       = $2\pi r^{2} + 2\pi rx\\ 2\pi r(r+rx)$