You have a wire that is 23 cm long. you wish to cut it into two pieces. one piece will be bent into the shape of a square. the other piece will be bent into the shape of a circle. let a represent the total area of the square and the circle. what is the circumference of the circle when a is a minimum?

In explanation you see the answer but it I see it send it  different  so I just added a file so you can see clearly the way how this exercise works. Please, see the file attached

Explanation:

Let x = circumference of the circle

then

(53-x) = the perimeter of the square

and \frac{53-\times}{4} = the side of the square

and

\left(\frac{53-\times}{4}\right)^2 = the area of the square

:

Find the area of the circle using the circumference

find the radius (r)

2*pi*r = x

r = \frac{\times}{2*\mathbit{\pi}}

r = \frac{\times}{6.28}

Find the area of the circle

\mathbit{A}=\pi r^2

Replace r with \frac{\times}{6.28}

A = π * \left(\frac{\times}{6.28}\right)^2=  * \left(\frac{\times^2}{39.48}\right)      

cancel pi into 39.48

A =  \left(\frac{\times^2}{12.566}\right)

Total area of circle and square

A =  \left(\frac{\times^2}{12.566}\right)+\left(\frac{53-\times}{4}\right)^2  = \left(\frac{\times^2}{12.566}\right)  + \left(\frac{{2809-106\times+\times}^2}{16}\right)

:

convert these fractions to decimal coefficients

A(x) = .0796x^2 + .0625x^2 - 6.625x + 175.5625

A(x) = .1421x^2 - 6.625x + 175.5625

:

Find the axis of symmetry of this quadratic equation (min area)

x =   \frac{-\left(-6.625\right)}{2*.1421}

x =   \frac{6.625}{.2842}

x = 23.31 cm is the circumference when they have min area

since there isn't any suggested answers i will give you the reason and hopefully you can figure it

south africa, or rather all of africa, has not been in the location in which it is nowadays throughout the geological past. after the super-continent pangaea started to break apart, gondwanaland, in which was africa, moved further south, and parts of it came to be very close to the south pole. because the climate in this part of the world is very cold due to the very dispersed sun rays in there, what is now south africa, started to freeze, and the conditions were right for the formation of glaciers. the glaciers existed up until the period when gondwanaland started to break apart and what is now africa started to move towards north, sp it warmed up and the glaciers melted.