You can
put this solution on YOUR website!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.
:
Square: Perimeter = x
1 side = x/4
Area =
:
Circle: Circumference = (100-x)
Find the radius (r)
2*pi*r = (100-x)
r =
:
Area = pi*r^2
:
Substitute
for r
:
Area =
:
Area =
; squared the radius
:
Area =
; canceled pi
:
Total area of square and circle:
A(x) =
+
:
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
