(1 point) A wire 5 meters long is cut into two pieces. One piece is bent into a

Question

(1 point) A wire 5 meters long is cut into two pieces. One piece is bent into a square for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures each: For the square: For the circle: (for both, include units) Where should the wire be cut to maximize the total area? Again, give the length of wire used for each: For the square: For the circle: (for both, include units)

Explanation / Answer

let x meters is used for the square

=>5-x meters is used for circle

let side of square =s ,rafius of circle =r

=>perimeter of square =4s =x ,circumference of circle=2r =5-x

=>s=(x/4), r=((5-x)/2)

total area,A=s2+r2

total area,A=(x/4)2+((5-x)/2)2

total area,A=(x2/16)+((5-x)2/4)

dA/dx=(x/8)-((5-x)/2)

for local extrema, dA/dx=0

(x/8)-((5-x)/2)=0

=>(x/8)=((5-x)/2)

=>x=(8(5-x)/2)

=>x=4(5-x)

=>x=20-4x

=>x+4x=20

=>x(4+)=20

=>x=20/(4+)

for x=0

A=(25/4)=1.9894

for x=5

A=(25/16)=1.5625

for x=20/(4+)

A=25/4(4+)=0.87515

minimum total area:

for the square :20/(4+)=2.8m

for the circle:5-(20/(4+)) =5/(4+) =2.2 m

maxmum total area:

for the square :0 m

for the circle:5 m

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