A wire of length 12 inches is divided into two pieces and the pieces are bent in
ID: 3079247 • Letter: A
Question
A wire of length 12 inches is divided into two pieces and the pieces are bent into a square and a circle. a.) Let Ls be the length of the wire formed into the square and Lc be the length of the wire formed into the circle. Write equations for the area of the square and circle, As and Ac in terms of Ls and Lc, respectively. i. As= ii Ac= b.) Create a table of values for As and Ac for various cuts of the wire. For example, if the wire is cut so that Ls=1, then Lc=11. Then use these values to calculate As and Ac. Pick at lease 7 different cuts. c.)Observe and describe what happens to the sum of the areas of the two shapes. Please Explain! Thank you!Explanation / Answer
A)
The area of a circle is just the square of the side length.
In this case the length of each side will be Ls/4
As = (Ls/4)^2 = Ls^2 / 16
The area of a circle is pi*r^2.
The circumference of a circle is 2*pi*r.
So r = (circumference)/(2*pi), we know the circumference is Lc.
r = Lc/(2*pi)
Ac = pi*r^2 = pi*(Lc/(2*pi))^2 = Lc^2 / (4*pi)
B)
C)
The total area gets smaller as the lengths Ls and Lc become closer in size, then starts to get bigger again as Ls gets bigger and Lc gets smaller.
If we graphed the function for total area of the shapes in relation to one to one of the Lengths, we would se a local minimum somewhere between Ls = 6 and Ls=8.
Ls Lc As Ac Total Area 1 11 0.0625 9.628882 9.691382 2 10 0.25 7.957754 8.207754 3 9 0.5625 6.445781 7.008281 4 8 1 5.092962 6.092962 5 7 1.5625 3.899299 5.461799 6 6 2.25 2.864791 5.114791 7 5 3.0625 1.989438 5.051938 8 4 4 1.273241 5.273241 9 3 5.0625 0.716198 5.778698 10 2 6.25 0.31831 6.56831 11 1 7.5625 0.079578 7.642078Related Questions
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