Apiece of wire of length 56 is cut, and the resulting two pieces are formed to make a circle and a square. where should the wire be cut to (a) minimize and (b) maximize the combined area of the circle and the square?

1) Cut the wire at a distance of 31.36 and draw this length into a square and the remaining into a circle to minimize the area.

2) To maximize the area do not cut the wire but make the whole wire into a circle.

Step-by-step explanation:

let the wire be cut at a distance of x

We make a square of this wire

For the remaining length of (56 - x) we make a circle

Thus

Area of square =

Similarly the area of the circle equals

Thus summing the areas we get

1) to find the critical points we differentiate the given area with respect to 'x'

Thus we have

Thus the length of the square wire should be x = 31.36

The length of the circular portion should be 56 - 31.36 = 24.64.

These lengths shall give the minimum combined area of the 2 figures.

2) Since the given function is a quadratic thus with the graph attached below we can see the maximum area occurs if all the wire is to be made into a circle thus to maximize the area the wire shall not be cut and wholly shaped into a circle.

associative property of multiplication

the transverse axis of a hyperbola is the line that contains the two vertices and the two focuses.

in this example (hyperbola of equation x^2/2 - y^2/4 = 1), the transverse axis is the  x-axis  . 

step-by-step explanation: