9514 1404 393
Answer:
x = 1/2
Step-by-step explanation:
There are some different ways to find the area of a donut shape. One is to multiply its centerline length by its width. We'll use that here to find the shaded area of the sector.
The centerline radius of shaded area A is ...
(6x -2) cm + (1/2)(4x cm) = 8x -2 cm
The length of the centerline is ...
s = rθ = (8x -2)(π/4) . . . . . . for a central angle of 45° = π/4 radians
Then the area A is ...
area A = (π/4)(8x -2)(4x) = 2πx(4x -1) . . . cm^2
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The radius of circle B is ((4 -4x) cm)/2 = (2 -2x) cm. The area of circle B is ...
A = πr^2 = π(2 -2x)^2 cm^2
We're told the two areas are equal, so we have ...
2πx(4x -1) = π(2 -2x)^2 . . . . . . . expressions for the areas are equal
8x^2 -2x = 4x^2 -8x +4 . . . . . . divide by π, eliminate parentheses
4x^2 +6x -4 = 0 . . . . . . . . . . . . subtract the right-side expression
2(2x -1)(x +2) = 0 . . . . . . . . . . . factor
x = 1/2 or -2 . . . . . . . only the positive solution is useful here
The value of x is 1/2.
