SOLUTION: 3)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

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Question 78242: 3)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.
(a) Express as a function of x, the sum A of the area of the square and the circle.
(b) Find the value of x, if any, that makes the area of the square equal to the area of the circle.
(c) Specify the domain and the range of the function A(x)
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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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.
:
(a) Express as a function of x, the sum A of the area of the square and the circle.
:
Square: Perimeter = x
1 side = x/4
Area = %28x%2F4%29%5E2
:
Circle: Circumference = (100-x)
Find the radius (r)
2*pi*r = (100-x)
r = %28%28100-x%29%29%2F%28%282pi%29%29
:
Area = pi*r^2
:
Substitute %28%28100-x%29%29%2F%28%282pi%29%29 for r
:
Area = pi%28%28%28100-x%29%29%2F%28%282pi%29%29%29%5E2
:
Area = pi%2810000-200x%2Bx%5E2%29%2F%284pi%5E2%29; squared the radius
:
Area = %2810000-200x%2Bx%5E2%29%2F%284pi%29; canceled pi
:
Total area of square and circle:
A(x) = %28x%2F4%29%5E2 + %28x%5E2-200x%2B10000%29%2F%284pi%29
:
A graph of this would be interesting:

:
:
(b) Find the value of x, if any, that makes the area of the square equal to the area of the circle.
:
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)
;

:
It looks the the areas are equal when x = 53, which is the minimum in the first graph.
:
:
(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
:
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
:
Notice that 2*175.5 = 351 which is about the total area shown on the 1st graph at x=53
:
I know it did not say anything about graphing but it makes more sense when you can see it. Hope this helped