Thread: calculus for areas

Hi:
I have this question that follows. It is in a differentiation section of the book. It asks to find area and value of x. But I thought when I differentiate it is to find the gradient of a tangent to a curve. What don't I know here?

The question is as follows:
A piece of wire of total length 12m is cut into two pieces. One piece of wite is bent into a rectangle of sides xm and 3xm and the other is bent to form the boundary of a square. Show that the total area enclosed by the rectangle and the square is given by:



Find the value of x if the total area enclosed is a minimum. Verify that your stationary value is a minimum.

Thank you

Right well lets first find perimeters.

for the rectangle its .

So for the square it's just whatever is left which is .

So area of the rectangle is width times length which is .

Each side of the square has length .

Hence area of the square is...



Hence total area is...

.

To find the minimum find the derivative and set it to be 0.

To show it is a minimum find the second derivative and check if it is greater than 0.

EDIT: I see Soroban is replying to this thread as well. He will explain this better if you are still confused.

ok. I have worked the question based upon Deadstar's workings and got the following:

Hello, stealthmaths!

A piece of wire of total length 12 m is cut into two pieces.
One piece of wire is bent into a rectangle of sides m and m
and the other is bent to form the boundary of a square.

Show that the total area enclosed by the rectangle and the square
is given by: .

Find the value of if the total area enclosed is a minimum.
Verify that your stationary value is a minimum.

The 12-meter wire is cut into two pieces of length and .

Code:

      : - - - - -  12 - - - - - :
      *---------*---------------*
      : -  a  - : - -  12-a - - :

The left piece of wire is bent into a

Code:

            3x
      * - - - - - *
      |           |
    x |           | x
      |           |
      * - - - - - *
           3x

The area of this rectangle is: .



We have: .

The right piece of wire has length: .
It is bent into a square.

Code:

          s
      * - - - *
      |       |
    s |       | s
      |       |
      * - - - *
          s

We have: .

The area of this square is: .


Therefore, the total area is: .