Question 78242
A piece of wire 100 cm long is cut into two pieces of length x and 100-x, respectively. the first piece is bent into the shape of a square, and the second is bent into the shape of a circle.
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(a) Express as a function of x, the sum A of the area of the square and the circle.
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Square: Perimeter = x
1 side = x/4
Area = {{{(x/4)^2}}}
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Circle: Circumference = (100-x)
Find the radius (r)
2*pi*r = (100-x)
r = {{{((100-x))/((2pi))}}}
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Area = pi*r^2
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Substitute {{{((100-x))/((2pi))}}} for r
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Area = {{{pi(((100-x))/((2pi)))^2}}}
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Area = {{{pi(10000-200x+x^2)/(4pi^2)}}}; squared the radius
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Area = {{{(10000-200x+x^2)/(4pi)}}}; canceled pi
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Total area of square and circle:
A(x) = {{{(x/4)^2}}} + {{{(x^2-200x+10000)/(4pi)}}}
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A graph of this would be interesting:
{{{ graph( 300, 200, -20, 110, -100, 800, (x/4)^2+(x^2 - 200x + 10000)/12.566 ) }}}
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(b) Find the value of x, if any, that makes the area of the square equal to the area of the circle.
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A graph would help here, 
Let y = (x/4)^2 be the purple line (area of the square) y
and
Let y = (x^2 - 200x + 10000)/(4pi) be the green line (area of the circle)
;
{{{ graph( 300, 200, -20, 110, -100, 800, (x/4)^2, (x^2 - 200x + 10000)/12.566 ) }}}
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It looks the the areas are equal when x = 53, which is the minimum in the first graph. 
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(c) Specify the domain and the range of the function A(x)
We know the domain has to be from 0 to 100
Using the first graph it appears that the range >350
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Check our to see if x = 53 gives equal areas
(53/4)^2 = 175.56 sq cm
(53^2 - 200(53) + 10000 divided by 4pi = 175.78, close enough
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Notice that 2*175.5 = 351 which is about the total area shown on the 1st graph at x=53
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I know it did not say anything about graphing but it makes more sense when you can see it. Hope this helped