piece of wire 8 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much wire should be used for the square in order to maximize the total area

piece of wire 8 m long

one piece is bent into square:

the square has four equal sides , so at least 4 m has to be cut from the wire to form a square with side=1 m.

the perimeter of the square =2L+2W=2(1)+2(1)=4 m

that is the max. amount can be cut from the wire, since the other part is bent into a circle.

( note if you cut more, the square will take the whole wire)

Perimeter=2L+2W=2(2)=2(2)=8 m and the area=2*2=4 m²)

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240