Steve likes to entertain friends at parties with wire tricks

Steve likes to entertain friends at parties with \"wire tricks.\" Suppose he takes a piece of wire 68 inches long and cuts it into two pieces. Steve takes the first piece of wire and bends it into the shape of a perfect circle. He then proceeds to bend the second piece of wire into the shape of a perfect square. What should the lengths of the wires be so that the total area of the circle and square combined is as small as possible? (Round your answers to two decimal places.)

Solution

circumference of circle + perimeter of the square = total wire used.(68in)

the question at the end of the paragraph asks what the length of the wires is to make the combined area(of circle and square) as small as possible

suppose for a second here that yew\'re going to use 64 inches for your square. 16 inch sides. so the area of that very large square is 256in^2... there is 4 inches left to build a perfect circle. so the circumference is 4. circumference is pi times the diameter. 4/pi =1.27. diameter 1.28/2 =radius=0.64. area of a circle =pi * r^2. 0.64^2=1.27.

so your area is 1.27 for the circle and 256 for the square making a total of 257.27 ishhhhh

your goal is to make it the smallest total possible.

what we want to know first is which shape will make the LEAST area by using the most wire? we want to know because then most of our area will come from that shape which is using most of the wire and we will have a smaller total area.

let\'s use an easy number to manipulate and figure out which one has a smaller area if they both use the same amount of wire

let\'s say they both used 100 inches of wire( i know it\'s more than the problem has but it\'s easy to manipulate to teach)

the square would have a side of 25 in and an area of 625in^2
the circle would have a circumference of 100, a diameter of 31.83, radius of 15.92 and area of 795.77

obviously if they both used the same amount of wire, a circle would have more area so we\'ll use as much wire as we can on the square.

we kind of already did this by using 64/68 inches on the square. assuming steve didn\'t make an infinitely small circle and use alllll of the rest on the square, i\'ll say he made a PERFECT square using the biggest whole numbers he could for it (16 inch side) and bending the rest into a circle.

length of wire for square 64inches
length of wire for circle 4inches
combined minimal area(assuming no limits) roughly 257.27in^2